Polytope of Type {2,4,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,4,4}*64
if this polytope has a name.
Group : SmallGroup(64,202)
Rank : 4
Schlafli Type : {2,4,4}
Number of vertices, edges, etc : 2, 4, 8, 4
Order of s₀s₁s₂s₃ : 4
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,4,4,2} of size 128
   {2,4,4,4} of size 256
   {2,4,4,6} of size 384
   {2,4,4,3} of size 384
   {2,4,4,8} of size 512
   {2,4,4,8} of size 512
   {2,4,4,4} of size 512
   {2,4,4,6} of size 576
   {2,4,4,10} of size 640
   {2,4,4,12} of size 768
   {2,4,4,6} of size 768
   {2,4,4,14} of size 896
   {2,4,4,5} of size 960
   {2,4,4,18} of size 1152
   {2,4,4,6} of size 1152
   {2,4,4,4} of size 1152
   {2,4,4,9} of size 1152
   {2,4,4,20} of size 1280
   {2,4,4,22} of size 1408
   {2,4,4,10} of size 1600
   {2,4,4,26} of size 1664
   {2,4,4,6} of size 1728
   {2,4,4,28} of size 1792
   {2,4,4,30} of size 1920
   {2,4,4,15} of size 1920
   {2,4,4,5} of size 1920
   {2,4,4,10} of size 1920
   {2,4,4,10} of size 1920
   {2,4,4,6} of size 1920
Vertex Figure Of :
   {2,2,4,4} of size 128
   {3,2,4,4} of size 192
   {4,2,4,4} of size 256
   {5,2,4,4} of size 320
   {6,2,4,4} of size 384
   {7,2,4,4} of size 448
   {9,2,4,4} of size 576
   {10,2,4,4} of size 640
   {11,2,4,4} of size 704
   {12,2,4,4} of size 768
   {13,2,4,4} of size 832
   {14,2,4,4} of size 896
   {15,2,4,4} of size 960
   {17,2,4,4} of size 1088
   {18,2,4,4} of size 1152
   {19,2,4,4} of size 1216
   {20,2,4,4} of size 1280
   {21,2,4,4} of size 1344
   {22,2,4,4} of size 1408
   {23,2,4,4} of size 1472
   {25,2,4,4} of size 1600
   {26,2,4,4} of size 1664
   {27,2,4,4} of size 1728
   {28,2,4,4} of size 1792
   {29,2,4,4} of size 1856
   {30,2,4,4} of size 1920
   {31,2,4,4} of size 1984
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,2,4}*32, {2,4,2}*32
   4-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,4,4}*128, {2,4,8}*128a, {2,8,4}*128a, {2,4,8}*128b, {2,8,4}*128b, {2,4,4}*128
   3-fold covers : {2,4,12}*192a, {2,12,4}*192a, {6,4,4}*192
   4-fold covers : {2,4,8}*256a, {2,8,4}*256a, {2,8,8}*256a, {2,8,8}*256b, {2,8,8}*256c, {2,8,8}*256d, {4,4,8}*256a, {8,4,4}*256a, {4,4,8}*256b, {8,4,4}*256b, {4,8,4}*256a, {4,4,4}*256a, {4,4,4}*256b, {4,8,4}*256b, {4,8,4}*256c, {4,8,4}*256d, {2,4,16}*256a, {2,16,4}*256a, {2,4,16}*256b, {2,16,4}*256b, {2,4,4}*256, {2,4,8}*256b, {2,8,4}*256b
   5-fold covers : {2,4,20}*320, {2,20,4}*320, {10,4,4}*320
   6-fold covers : {4,12,4}*384a, {4,4,12}*384, {12,4,4}*384, {2,4,24}*384a, {2,24,4}*384a, {2,4,12}*384a, {2,12,4}*384a, {2,4,24}*384b, {2,24,4}*384b, {2,8,12}*384a, {2,12,8}*384a, {2,8,12}*384b, {2,12,8}*384b, {6,4,8}*384a, {6,8,4}*384a, {6,4,8}*384b, {6,8,4}*384b, {6,4,4}*384a
   7-fold covers : {2,4,28}*448, {2,28,4}*448, {14,4,4}*448
   8-fold covers : {2,8,8}*512a, {8,4,8}*512a, {8,4,8}*512b, {4,4,4}*512a, {4,8,8}*512a, {8,8,4}*512a, {4,8,8}*512b, {8,8,4}*512b, {4,4,8}*512a, {8,4,4}*512a, {4,8,8}*512c, {8,8,4}*512c, {4,8,8}*512d, {8,8,4}*512d, {4,8,8}*512e, {4,8,8}*512f, {8,8,4}*512e, {8,8,4}*512f, {4,8,8}*512g, {8,8,4}*512g, {4,8,8}*512h, {8,8,4}*512h, {4,4,8}*512b, {8,4,4}*512b, {4,4,8}*512c, {8,4,4}*512c, {4,8,4}*512a, {4,8,4}*512b, {4,8,4}*512c, {4,8,4}*512d, {8,4,8}*512c, {8,4,8}*512d, {2,4,8}*512a, {2,8,4}*512a, {2,8,8}*512b, {2,8,8}*512c, {2,8,8}*512d, {2,4,16}*512a, {2,16,4}*512a, {2,4,16}*512b, {2,16,4}*512b, {2,8,16}*512a, {2,16,8}*512a, {2,8,16}*512b, {2,16,8}*512b, {2,8,16}*512c, {2,8,16}*512d, {2,16,8}*512c, {2,16,8}*512d, {2,8,16}*512e, {2,8,16}*512f, {2,16,8}*512e, {2,16,8}*512f, {4,4,16}*512a, {16,4,4}*512a, {4,4,16}*512b, {16,4,4}*512b, {4,4,4}*512b, {4,4,4}*512c, {4,8,4}*512e, {4,8,4}*512f, {4,8,4}*512g, {4,8,4}*512h, {4,4,8}*512d, {8,4,4}*512d, {4,16,4}*512a, {4,16,4}*512b, {4,16,4}*512c, {4,16,4}*512d, {2,4,32}*512a, {2,32,4}*512a, {2,4,32}*512b, {2,32,4}*512b, {2,4,4}*512, {2,4,8}*512b, {2,8,4}*512b, {2,4,8}*512c, {2,4,8}*512d, {2,8,4}*512c, {2,8,4}*512d, {2,8,8}*512e, {2,8,8}*512f, {2,8,8}*512g, {2,8,8}*512h
   9-fold covers : {2,4,36}*576a, {2,36,4}*576a, {18,4,4}*576, {6,4,12}*576, {6,12,4}*576a, {6,12,4}*576b, {2,12,12}*576a, {2,12,12}*576b, {2,12,12}*576c, {6,12,4}*576c, {2,4,4}*576, {6,4,4}*576, {2,4,12}*576, {2,12,4}*576
   10-fold covers : {4,20,4}*640, {4,4,20}*640, {20,4,4}*640, {2,4,40}*640a, {2,40,4}*640a, {2,4,20}*640, {2,20,4}*640, {2,4,40}*640b, {2,40,4}*640b, {2,8,20}*640a, {2,20,8}*640a, {2,8,20}*640b, {2,20,8}*640b, {10,4,8}*640a, {10,8,4}*640a, {10,4,8}*640b, {10,8,4}*640b, {10,4,4}*640
   11-fold covers : {2,4,44}*704, {2,44,4}*704, {22,4,4}*704
   12-fold covers : {6,4,8}*768a, {6,8,4}*768a, {2,8,12}*768a, {2,12,8}*768a, {2,4,24}*768a, {2,24,4}*768a, {6,8,8}*768a, {6,8,8}*768b, {6,8,8}*768c, {2,8,24}*768a, {2,24,8}*768a, {2,8,24}*768b, {2,8,24}*768c, {2,24,8}*768b, {2,24,8}*768c, {6,8,8}*768d, {2,8,24}*768d, {2,24,8}*768d, {8,4,12}*768a, {12,4,8}*768a, {4,12,8}*768a, {8,12,4}*768a, {4,4,24}*768a, {24,4,4}*768a, {8,4,12}*768b, {12,4,8}*768b, {4,12,8}*768b, {8,12,4}*768b, {4,4,24}*768b, {24,4,4}*768b, {4,8,12}*768a, {12,8,4}*768a, {4,24,4}*768a, {4,4,12}*768a, {12,4,4}*768a, {4,12,4}*768a, {4,12,4}*768b, {4,4,12}*768b, {12,4,4}*768b, {4,8,12}*768b, {12,8,4}*768b, {4,24,4}*768b, {4,24,4}*768c, {4,8,12}*768c, {12,8,4}*768c, {4,8,12}*768d, {12,8,4}*768d, {4,24,4}*768d, {6,4,16}*768a, {6,16,4}*768a, {2,12,16}*768a, {2,16,12}*768a, {2,4,48}*768a, {2,48,4}*768a, {6,4,16}*768b, {6,16,4}*768b, {2,12,16}*768b, {2,16,12}*768b, {2,4,48}*768b, {2,48,4}*768b, {6,4,4}*768a, {6,4,8}*768b, {6,8,4}*768b, {2,4,12}*768a, {2,4,24}*768b, {2,12,4}*768a, {2,24,4}*768b, {2,8,12}*768b, {2,12,8}*768b, {2,4,12}*768d, {4,12,4}*768f, {2,12,4}*768d, {2,12,12}*768a, {6,4,4}*768e, {6,12,4}*768a
   13-fold covers : {2,4,52}*832, {2,52,4}*832, {26,4,4}*832
   14-fold covers : {4,28,4}*896, {4,4,28}*896, {28,4,4}*896, {2,4,56}*896a, {2,56,4}*896a, {2,4,28}*896, {2,28,4}*896, {2,4,56}*896b, {2,56,4}*896b, {2,8,28}*896a, {2,28,8}*896a, {2,8,28}*896b, {2,28,8}*896b, {14,4,8}*896a, {14,8,4}*896a, {14,4,8}*896b, {14,8,4}*896b, {14,4,4}*896
   15-fold covers : {10,4,12}*960, {10,12,4}*960a, {6,4,20}*960, {6,20,4}*960, {2,12,20}*960, {2,20,12}*960, {2,4,60}*960a, {2,60,4}*960a, {30,4,4}*960
   17-fold covers : {34,4,4}*1088, {2,4,68}*1088, {2,68,4}*1088
   18-fold covers : {4,4,36}*1152, {36,4,4}*1152, {4,36,4}*1152a, {4,12,12}*1152a, {4,12,12}*1152b, {12,12,4}*1152a, {12,12,4}*1152b, {4,12,12}*1152c, {12,12,4}*1152c, {12,4,12}*1152, {4,4,4}*1152a, {4,4,4}*1152b, {4,12,4}*1152a, {4,12,4}*1152b, {4,4,12}*1152, {12,4,4}*1152, {18,4,8}*1152a, {18,8,4}*1152a, {2,8,36}*1152a, {2,36,8}*1152a, {2,4,72}*1152a, {2,72,4}*1152a, {6,8,12}*1152a, {6,12,8}*1152a, {6,12,8}*1152b, {6,12,8}*1152c, {6,24,4}*1152a, {6,4,24}*1152a, {6,24,4}*1152b, {6,24,4}*1152c, {2,12,24}*1152a, {2,12,24}*1152b, {2,24,12}*1152a, {2,24,12}*1152b, {2,12,24}*1152c, {2,24,12}*1152c, {6,4,8}*1152a, {2,4,8}*1152a, {2,4,24}*1152a, {2,8,4}*1152a, {2,24,4}*1152a, {2,8,12}*1152a, {2,12,8}*1152a, {6,8,4}*1152a, {18,4,8}*1152b, {18,8,4}*1152b, {2,8,36}*1152b, {2,36,8}*1152b, {2,4,72}*1152b, {2,72,4}*1152b, {6,8,12}*1152b, {6,12,8}*1152d, {6,12,8}*1152e, {6,12,8}*1152f, {6,24,4}*1152d, {6,4,24}*1152b, {6,24,4}*1152e, {6,24,4}*1152f, {2,12,24}*1152d, {2,12,24}*1152e, {2,24,12}*1152d, {2,24,12}*1152e, {2,12,24}*1152f, {2,24,12}*1152f, {2,4,8}*1152b, {2,4,24}*1152b, {2,8,4}*1152b, {2,24,4}*1152b, {6,4,8}*1152b, {2,8,12}*1152b, {2,12,8}*1152b, {6,8,4}*1152b, {18,4,4}*1152a, {2,4,36}*1152a, {2,36,4}*1152a, {6,4,12}*1152a, {6,12,4}*1152a, {6,12,4}*1152b, {6,12,4}*1152c, {2,12,12}*1152a, {2,12,12}*1152b, {2,12,12}*1152c, {2,4,4}*1152, {2,4,12}*1152, {2,12,4}*1152, {6,4,4}*1152a
   19-fold covers : {38,4,4}*1216, {2,4,76}*1216, {2,76,4}*1216
   20-fold covers : {10,4,8}*1280a, {10,8,4}*1280a, {2,8,20}*1280a, {2,20,8}*1280a, {2,4,40}*1280a, {2,40,4}*1280a, {10,8,8}*1280a, {10,8,8}*1280b, {10,8,8}*1280c, {2,8,40}*1280a, {2,40,8}*1280a, {2,8,40}*1280b, {2,8,40}*1280c, {2,40,8}*1280b, {2,40,8}*1280c, {10,8,8}*1280d, {2,8,40}*1280d, {2,40,8}*1280d, {8,4,20}*1280a, {20,4,8}*1280a, {4,20,8}*1280a, {8,20,4}*1280a, {4,4,40}*1280a, {40,4,4}*1280a, {8,4,20}*1280b, {20,4,8}*1280b, {4,20,8}*1280b, {8,20,4}*1280b, {4,4,40}*1280b, {40,4,4}*1280b, {4,8,20}*1280a, {20,8,4}*1280a, {4,40,4}*1280a, {4,4,20}*1280a, {20,4,4}*1280a, {4,20,4}*1280a, {4,20,4}*1280b, {4,4,20}*1280b, {20,4,4}*1280b, {4,8,20}*1280b, {20,8,4}*1280b, {4,40,4}*1280b, {4,40,4}*1280c, {4,8,20}*1280c, {20,8,4}*1280c, {4,8,20}*1280d, {20,8,4}*1280d, {4,40,4}*1280d, {10,4,16}*1280a, {10,16,4}*1280a, {2,16,20}*1280a, {2,20,16}*1280a, {2,4,80}*1280a, {2,80,4}*1280a, {10,4,16}*1280b, {10,16,4}*1280b, {2,16,20}*1280b, {2,20,16}*1280b, {2,4,80}*1280b, {2,80,4}*1280b, {10,4,4}*1280, {10,4,8}*1280b, {10,8,4}*1280b, {2,4,20}*1280a, {2,4,40}*1280b, {2,20,4}*1280a, {2,40,4}*1280b, {2,8,20}*1280b, {2,20,8}*1280b
   21-fold covers : {14,4,12}*1344, {14,12,4}*1344a, {6,4,28}*1344, {6,28,4}*1344, {2,12,28}*1344, {2,28,12}*1344, {2,4,84}*1344a, {2,84,4}*1344a, {42,4,4}*1344
   22-fold covers : {4,4,44}*1408, {44,4,4}*1408, {4,44,4}*1408, {22,4,8}*1408a, {22,8,4}*1408a, {2,8,44}*1408a, {2,44,8}*1408a, {2,4,88}*1408a, {2,88,4}*1408a, {22,4,8}*1408b, {22,8,4}*1408b, {2,8,44}*1408b, {2,44,8}*1408b, {2,4,88}*1408b, {2,88,4}*1408b, {22,4,4}*1408, {2,4,44}*1408, {2,44,4}*1408
   23-fold covers : {46,4,4}*1472, {2,4,92}*1472, {2,92,4}*1472
   25-fold covers : {2,4,100}*1600, {2,100,4}*1600, {50,4,4}*1600, {10,4,20}*1600, {10,20,4}*1600a, {10,20,4}*1600b, {2,20,20}*1600a, {2,20,20}*1600b, {2,20,20}*1600c, {10,20,4}*1600c, {2,4,4}*1600, {10,4,4}*1600, {2,4,20}*1600, {2,20,4}*1600
   26-fold covers : {4,4,52}*1664, {52,4,4}*1664, {4,52,4}*1664, {26,4,8}*1664a, {26,8,4}*1664a, {2,8,52}*1664a, {2,52,8}*1664a, {2,4,104}*1664a, {2,104,4}*1664a, {26,4,8}*1664b, {26,8,4}*1664b, {2,8,52}*1664b, {2,52,8}*1664b, {2,4,104}*1664b, {2,104,4}*1664b, {26,4,4}*1664, {2,4,52}*1664, {2,52,4}*1664
   27-fold covers : {2,4,108}*1728a, {2,108,4}*1728a, {54,4,4}*1728, {18,4,12}*1728, {18,12,4}*1728a, {6,4,36}*1728, {6,36,4}*1728a, {6,36,4}*1728b, {6,12,4}*1728a, {6,12,4}*1728b, {6,12,12}*1728a, {2,12,36}*1728a, {2,12,36}*1728b, {2,36,12}*1728a, {2,36,12}*1728b, {2,12,12}*1728a, {2,12,12}*1728b, {2,12,12}*1728c, {18,12,4}*1728b, {6,12,4}*1728c, {2,4,12}*1728a, {2,4,12}*1728b, {6,4,4}*1728a, {6,12,4}*1728h, {6,12,4}*1728i, {2,12,4}*1728a, {2,12,4}*1728b, {2,12,12}*1728d, {2,12,12}*1728e, {2,12,12}*1728f, {2,12,12}*1728g, {6,12,12}*1728b, {6,12,12}*1728c, {6,12,12}*1728d, {6,12,12}*1728e, {6,12,12}*1728f, {2,12,12}*1728h, {6,12,4}*1728j, {6,12,12}*1728g, {6,4,12}*1728a, {2,4,12}*1728c, {2,4,12}*1728d, {2,12,4}*1728c, {2,12,4}*1728d, {2,12,12}*1728i, {2,12,12}*1728j, {6,4,4}*1728b, {6,4,4}*1728c, {6,4,12}*1728b, {6,12,4}*1728n, {6,12,4}*1728o, {6,12,4}*1728p, {6,12,4}*1728q, {2,12,12}*1728k, {2,12,12}*1728l
   28-fold covers : {14,4,8}*1792a, {14,8,4}*1792a, {2,8,28}*1792a, {2,28,8}*1792a, {2,4,56}*1792a, {2,56,4}*1792a, {14,8,8}*1792a, {14,8,8}*1792b, {14,8,8}*1792c, {2,8,56}*1792a, {2,56,8}*1792a, {2,8,56}*1792b, {2,8,56}*1792c, {2,56,8}*1792b, {2,56,8}*1792c, {14,8,8}*1792d, {2,8,56}*1792d, {2,56,8}*1792d, {8,4,28}*1792a, {28,4,8}*1792a, {4,28,8}*1792a, {8,28,4}*1792a, {4,4,56}*1792a, {56,4,4}*1792a, {8,4,28}*1792b, {28,4,8}*1792b, {4,28,8}*1792b, {8,28,4}*1792b, {4,4,56}*1792b, {56,4,4}*1792b, {4,8,28}*1792a, {28,8,4}*1792a, {4,56,4}*1792a, {4,4,28}*1792a, {28,4,4}*1792a, {4,28,4}*1792a, {4,28,4}*1792b, {4,4,28}*1792b, {28,4,4}*1792b, {4,8,28}*1792b, {28,8,4}*1792b, {4,56,4}*1792b, {4,56,4}*1792c, {4,8,28}*1792c, {28,8,4}*1792c, {4,8,28}*1792d, {28,8,4}*1792d, {4,56,4}*1792d, {14,4,16}*1792a, {14,16,4}*1792a, {2,16,28}*1792a, {2,28,16}*1792a, {2,4,112}*1792a, {2,112,4}*1792a, {14,4,16}*1792b, {14,16,4}*1792b, {2,16,28}*1792b, {2,28,16}*1792b, {2,4,112}*1792b, {2,112,4}*1792b, {14,4,4}*1792, {14,4,8}*1792b, {14,8,4}*1792b, {2,4,28}*1792, {2,4,56}*1792b, {2,28,4}*1792, {2,56,4}*1792b, {2,8,28}*1792b, {2,28,8}*1792b
   29-fold covers : {58,4,4}*1856, {2,4,116}*1856, {2,116,4}*1856
   30-fold covers : {4,4,60}*1920, {60,4,4}*1920, {4,60,4}*1920a, {4,20,12}*1920, {12,20,4}*1920, {4,12,20}*1920a, {20,12,4}*1920a, {12,4,20}*1920, {20,4,12}*1920, {30,4,8}*1920a, {30,8,4}*1920a, {2,8,60}*1920a, {2,60,8}*1920a, {2,4,120}*1920a, {2,120,4}*1920a, {10,8,12}*1920a, {10,12,8}*1920a, {6,8,20}*1920a, {6,20,8}*1920a, {10,4,24}*1920a, {10,24,4}*1920a, {6,4,40}*1920a, {6,40,4}*1920a, {2,12,40}*1920a, {2,40,12}*1920a, {2,20,24}*1920a, {2,24,20}*1920a, {30,4,8}*1920b, {30,8,4}*1920b, {2,8,60}*1920b, {2,60,8}*1920b, {2,4,120}*1920b, {2,120,4}*1920b, {10,8,12}*1920b, {10,12,8}*1920b, {6,8,20}*1920b, {6,20,8}*1920b, {10,4,24}*1920b, {10,24,4}*1920b, {6,4,40}*1920b, {6,40,4}*1920b, {2,12,40}*1920b, {2,40,12}*1920b, {2,20,24}*1920b, {2,24,20}*1920b, {30,4,4}*1920a, {2,4,60}*1920a, {2,60,4}*1920a, {10,4,12}*1920a, {10,12,4}*1920a, {6,4,20}*1920a, {6,20,4}*1920a, {2,12,20}*1920a, {2,20,12}*1920a
   31-fold covers : {62,4,4}*1984, {2,4,124}*1984, {2,124,4}*1984
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := (4,5)(6,8);;
s2 := ( 3, 4)( 5, 7)( 6, 9)( 8,10);;
s3 := (4,6)(5,8);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(10)!(1,2);
s1 := Sym(10)!(4,5)(6,8);
s2 := Sym(10)!( 3, 4)( 5, 7)( 6, 9)( 8,10);
s3 := Sym(10)!(4,6)(5,8);
poly := sub<Sym(10)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3 >;

to this polytope