Polytope of Type {8,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,10}*1280b
if this polytope has a name.
Group : SmallGroup(1280,323566)
Rank : 4
Schlafli Type : {8,4,10}
Number of vertices, edges, etc : 16, 32, 40, 10
Order of s₀s₁s₂s₃ : 20
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4,10}*640
   4-fold quotients : {4,4,10}*320
   5-fold quotients : {8,4,2}*256b
   8-fold quotients : {2,4,10}*160, {4,2,10}*160
   10-fold quotients : {4,4,2}*128
   16-fold quotients : {4,2,5}*80, {2,2,10}*80
   20-fold quotients : {4,4,2}*64
   32-fold quotients : {2,2,5}*40
   40-fold quotients : {2,4,2}*32, {4,2,2}*32
   80-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)(  8, 88)
(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)( 16, 96)
( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,116)( 22,117)( 23,118)( 24,119)
( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)( 32,107)
( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)( 40,105)
( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)( 48,128)
( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)( 56,136)
( 57,137)( 58,138)( 59,139)( 60,140)( 61,156)( 62,157)( 63,158)( 64,159)
( 65,160)( 66,151)( 67,152)( 68,153)( 69,154)( 70,155)( 71,146)( 72,147)
( 73,148)( 74,149)( 75,150)( 76,141)( 77,142)( 78,143)( 79,144)( 80,145);;
s1 := ( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)( 33, 38)
( 34, 39)( 35, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)( 71, 76)
( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81,101)( 82,102)( 83,103)( 84,104)
( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)( 92,117)
( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)(100,115)
(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)(128,148)
(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)(136,151)
(137,152)(138,153)(139,154)(140,155);;
s2 := (  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)(  8, 49)
(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)( 16, 56)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 76)( 22, 80)( 23, 79)( 24, 78)
( 25, 77)( 26, 71)( 27, 75)( 28, 74)( 29, 73)( 30, 72)( 31, 66)( 32, 70)
( 33, 69)( 34, 68)( 35, 67)( 36, 61)( 37, 65)( 38, 64)( 39, 63)( 40, 62)
( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)( 88,129)
( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)( 96,136)
( 97,140)( 98,139)( 99,138)(100,137)(101,156)(102,160)(103,159)(104,158)
(105,157)(106,151)(107,155)(108,154)(109,153)(110,152)(111,146)(112,150)
(113,149)(114,148)(115,147)(116,141)(117,145)(118,144)(119,143)(120,142);;
s3 := (  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)(  8, 46)
(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)( 16, 58)
( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 63)( 22, 62)( 23, 61)( 24, 65)
( 25, 64)( 26, 68)( 27, 67)( 28, 66)( 29, 70)( 30, 69)( 31, 73)( 32, 72)
( 33, 71)( 34, 75)( 35, 74)( 36, 78)( 37, 77)( 38, 76)( 39, 80)( 40, 79)
( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)( 88,126)
( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)( 96,138)
( 97,137)( 98,136)( 99,140)(100,139)(101,143)(102,142)(103,141)(104,145)
(105,144)(106,148)(107,147)(108,146)(109,150)(110,149)(111,153)(112,152)
(113,151)(114,155)(115,154)(116,158)(117,157)(118,156)(119,160)(120,159);;
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*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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(160)!(  1, 81)(  2, 82)(  3, 83)(  4, 84)(  5, 85)(  6, 86)(  7, 87)
(  8, 88)(  9, 89)( 10, 90)( 11, 91)( 12, 92)( 13, 93)( 14, 94)( 15, 95)
( 16, 96)( 17, 97)( 18, 98)( 19, 99)( 20,100)( 21,116)( 22,117)( 23,118)
( 24,119)( 25,120)( 26,111)( 27,112)( 28,113)( 29,114)( 30,115)( 31,106)
( 32,107)( 33,108)( 34,109)( 35,110)( 36,101)( 37,102)( 38,103)( 39,104)
( 40,105)( 41,121)( 42,122)( 43,123)( 44,124)( 45,125)( 46,126)( 47,127)
( 48,128)( 49,129)( 50,130)( 51,131)( 52,132)( 53,133)( 54,134)( 55,135)
( 56,136)( 57,137)( 58,138)( 59,139)( 60,140)( 61,156)( 62,157)( 63,158)
( 64,159)( 65,160)( 66,151)( 67,152)( 68,153)( 69,154)( 70,155)( 71,146)
( 72,147)( 73,148)( 74,149)( 75,150)( 76,141)( 77,142)( 78,143)( 79,144)
( 80,145);
s1 := Sym(160)!( 11, 16)( 12, 17)( 13, 18)( 14, 19)( 15, 20)( 31, 36)( 32, 37)
( 33, 38)( 34, 39)( 35, 40)( 51, 56)( 52, 57)( 53, 58)( 54, 59)( 55, 60)
( 71, 76)( 72, 77)( 73, 78)( 74, 79)( 75, 80)( 81,101)( 82,102)( 83,103)
( 84,104)( 85,105)( 86,106)( 87,107)( 88,108)( 89,109)( 90,110)( 91,116)
( 92,117)( 93,118)( 94,119)( 95,120)( 96,111)( 97,112)( 98,113)( 99,114)
(100,115)(121,141)(122,142)(123,143)(124,144)(125,145)(126,146)(127,147)
(128,148)(129,149)(130,150)(131,156)(132,157)(133,158)(134,159)(135,160)
(136,151)(137,152)(138,153)(139,154)(140,155);
s2 := Sym(160)!(  1, 41)(  2, 45)(  3, 44)(  4, 43)(  5, 42)(  6, 46)(  7, 50)
(  8, 49)(  9, 48)( 10, 47)( 11, 51)( 12, 55)( 13, 54)( 14, 53)( 15, 52)
( 16, 56)( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 76)( 22, 80)( 23, 79)
( 24, 78)( 25, 77)( 26, 71)( 27, 75)( 28, 74)( 29, 73)( 30, 72)( 31, 66)
( 32, 70)( 33, 69)( 34, 68)( 35, 67)( 36, 61)( 37, 65)( 38, 64)( 39, 63)
( 40, 62)( 81,121)( 82,125)( 83,124)( 84,123)( 85,122)( 86,126)( 87,130)
( 88,129)( 89,128)( 90,127)( 91,131)( 92,135)( 93,134)( 94,133)( 95,132)
( 96,136)( 97,140)( 98,139)( 99,138)(100,137)(101,156)(102,160)(103,159)
(104,158)(105,157)(106,151)(107,155)(108,154)(109,153)(110,152)(111,146)
(112,150)(113,149)(114,148)(115,147)(116,141)(117,145)(118,144)(119,143)
(120,142);
s3 := Sym(160)!(  1, 43)(  2, 42)(  3, 41)(  4, 45)(  5, 44)(  6, 48)(  7, 47)
(  8, 46)(  9, 50)( 10, 49)( 11, 53)( 12, 52)( 13, 51)( 14, 55)( 15, 54)
( 16, 58)( 17, 57)( 18, 56)( 19, 60)( 20, 59)( 21, 63)( 22, 62)( 23, 61)
( 24, 65)( 25, 64)( 26, 68)( 27, 67)( 28, 66)( 29, 70)( 30, 69)( 31, 73)
( 32, 72)( 33, 71)( 34, 75)( 35, 74)( 36, 78)( 37, 77)( 38, 76)( 39, 80)
( 40, 79)( 81,123)( 82,122)( 83,121)( 84,125)( 85,124)( 86,128)( 87,127)
( 88,126)( 89,130)( 90,129)( 91,133)( 92,132)( 93,131)( 94,135)( 95,134)
( 96,138)( 97,137)( 98,136)( 99,140)(100,139)(101,143)(102,142)(103,141)
(104,145)(105,144)(106,148)(107,147)(108,146)(109,150)(110,149)(111,153)
(112,152)(113,151)(114,155)(115,154)(116,158)(117,157)(118,156)(119,160)
(120,159);
poly := sub<Sym(160)|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*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*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
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