Polytope of Type {4,6}
Play with this polytope as a twisty puzzle
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6}*384a
if this polytope has a name.
Group : SmallGroup(384,17948)
Rank : 3
Schlafli Type : {4,6}
Number of vertices, edges, etc : 32, 96, 48
Order of s0s1s2 : 6
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Halving Operation
Facet Of :
{4,6,2} of size 768
{4,6,3} of size 1152
Vertex Figure Of :
{2,4,6} of size 768
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6}*192a
4-fold quotients : {4,6}*96
8-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
16-fold quotients : {4,3}*24, {2,6}*24
32-fold quotients : {2,3}*12
48-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {8,6}*768e, {8,6}*768g, {4,12}*768b, {4,6}*768a, {4,12}*768c, {8,6}*768m, {8,6}*768n, {4,6}*768b, {4,6}*768c, {4,12}*768g, {4,12}*768h
3-fold covers : {4,18}*1152a, {12,6}*1152b, {12,6}*1152c
5-fold covers : {20,6}*1920a, {4,30}*1920a
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2> of order 2.
28 facets:
20 of {4}*8
8 of {2}*4
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2> of order 2.
24 facets:
24 of {4}*8
20 vertex figures:
8 of {3}*6
12 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
24 facets:
24 of {4}*8
16 vertex figures:
16 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
14 facets:
10 of {4}*8
4 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s2*s1*s0*s1*s2*s1*s2, s1*s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s2*s1*s0*s2*s1*s0*s2*s1> of order 4.
12 facets:
12 of {4}*8
10 vertex figures:
4 of {3}*6
6 of {6}*12
P/N, where N=<s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s1> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
14 facets:
10 of {4}*8
4 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
18 facets:
6 of {4}*8
12 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
16 facets:
8 of {2}*4
8 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0> of order 4.
12 facets:
12 of {4}*8
12 vertex figures:
8 of {3}*6
4 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
14 facets:
10 of {4}*8
4 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
14 facets:
10 of {4}*8
4 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
12 facets:
12 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 4.
16 facets:
8 of {2}*4
8 of {4}*8
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
14 facets:
10 of {4}*8
4 of {2}*4
8 vertex figures:
8 of {6}*12
P/N, where N=<s0*s1*s0*s1, s1*s2*s1*s2*s1*s2> of order 8.
8 facets:
4 of {2}*4
4 of {4}*8
6 vertex figures:
4 of {3}*6
2 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
6 facets:
6 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 8.
8 facets:
4 of {2}*4
4 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
8 facets:
4 of {4}*8
4 of {2}*4
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
6 facets:
6 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 8.
6 facets:
6 of {4}*8
6 vertex figures:
4 of {3}*6
2 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
8 facets:
4 of {2}*4
4 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s1*s0*s1, s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1> of order 8.
10 facets:
8 of {2}*4
2 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 8.
6 facets:
6 of {4}*8
4 vertex figures:
4 of {6}*12
P/N, where N=<s1*s2*s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 8.
6 facets:
6 of {4}*8
6 vertex figures:
4 of {3}*6
2 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 8.
7 facets:
5 of {4}*8
2 of {2}*4
4 vertex figures:
4 of {6}*12
P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 8.
6 facets:
6 of {4}*8
4 vertex figures:
4 of {6}*12
Permutation Representation (GAP) :
s0 := ( 1,57)( 2,58)( 3,59)( 4,60)( 5,61)( 6,62)( 7,63)( 8,64)( 9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88);;
s1 := ( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48)(51,52)(53,60)(54,59)(55,57)(56,58)(61,62)(65,81)(66,82)(67,84)(68,83)(69,92)(70,91)(71,89)(72,90)(73,87)(74,88)(75,86)(76,85)(77,94)(78,93)(79,95)(80,96);;
s2 := ( 1,33)( 2,35)( 3,34)( 4,36)( 5,45)( 6,47)( 7,46)( 8,48)( 9,41)(10,43)(11,42)(12,44)(13,37)(14,39)(15,38)(16,40)(18,19)(21,29)(22,31)(23,30)(24,32)(26,27)(49,81)(50,83)(51,82)(52,84)(53,93)(54,95)(55,94)(56,96)(57,89)(58,91)(59,90)(60,92)(61,85)(62,87)(63,86)(64,88)(66,67)(69,77)(70,79)(71,78)(72,80)(74,75);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2");;
s0 := F.1;; s1 := F.2;; s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(96)!( 1,57)( 2,58)( 3,59)( 4,60)( 5,61)( 6,62)( 7,63)( 8,64)( 9,49)(10,50)(11,51)(12,52)(13,53)(14,54)(15,55)(16,56)(17,73)(18,74)(19,75)(20,76)(21,77)(22,78)(23,79)(24,80)(25,65)(26,66)(27,67)(28,68)(29,69)(30,70)(31,71)(32,72)(33,89)(34,90)(35,91)(36,92)(37,93)(38,94)(39,95)(40,96)(41,81)(42,82)(43,83)(44,84)(45,85)(46,86)(47,87)(48,88);
s1 := Sym(96)!( 3, 4)( 5,12)( 6,11)( 7, 9)( 8,10)(13,14)(17,33)(18,34)(19,36)(20,35)(21,44)(22,43)(23,41)(24,42)(25,39)(26,40)(27,38)(28,37)(29,46)(30,45)(31,47)(32,48)(51,52)(53,60)(54,59)(55,57)(56,58)(61,62)(65,81)(66,82)(67,84)(68,83)(69,92)(70,91)(71,89)(72,90)(73,87)(74,88)(75,86)(76,85)(77,94)(78,93)(79,95)(80,96);
s2 := Sym(96)!( 1,33)( 2,35)( 3,34)( 4,36)( 5,45)( 6,47)( 7,46)( 8,48)( 9,41)(10,43)(11,42)(12,44)(13,37)(14,39)(15,38)(16,40)(18,19)(21,29)(22,31)(23,30)(24,32)(26,27)(49,81)(50,83)(51,82)(52,84)(53,93)(54,95)(55,94)(56,96)(57,89)(58,91)(59,90)(60,92)(61,85)(62,87)(63,86)(64,88)(66,67)(69,77)(70,79)(71,78)(72,80)(74,75);
poly := sub<Sym(96)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope
Twisty Puzzle