Polytope of Type {4,4,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,10}*1280
Also Known As : {{4,4|4},{4,10|2}}. if this polytope has another name.
Group : SmallGroup(1280,323566)
Rank : 4
Schlafli Type : {4,4,10}
Number of vertices, edges, etc : 16, 32, 80, 10
Order of s₀s₁s₂s₃ : 40
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 : {4,4,2}*256
   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 := ( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)( 26, 31)( 27, 32)( 28, 33)
( 29, 34)( 30, 35)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 71)
( 67, 72)( 68, 73)( 69, 74)( 70, 75)(101,116)(102,117)(103,118)(104,119)
(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)(141,156)(142,157)
(143,158)(144,159)(145,160)(146,151)(147,152)(148,153)(149,154)(150,155);;
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 := (  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)(  8,129)
(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)( 16,136)
( 17,140)( 18,139)( 19,138)( 20,137)( 21,141)( 22,145)( 23,144)( 24,143)
( 25,142)( 26,146)( 27,150)( 28,149)( 29,148)( 30,147)( 31,151)( 32,155)
( 33,154)( 34,153)( 35,152)( 36,156)( 37,160)( 38,159)( 39,158)( 40,157)
( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 46, 86)( 47, 90)( 48, 89)
( 49, 88)( 50, 87)( 51, 91)( 52, 95)( 53, 94)( 54, 93)( 55, 92)( 56, 96)
( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61,101)( 62,105)( 63,104)( 64,103)
( 65,102)( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,111)( 72,115)
( 73,114)( 74,113)( 75,112)( 76,116)( 77,120)( 78,119)( 79,118)( 80,117);;
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, s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*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)!( 21, 36)( 22, 37)( 23, 38)( 24, 39)( 25, 40)( 26, 31)( 27, 32)
( 28, 33)( 29, 34)( 30, 35)( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)
( 66, 71)( 67, 72)( 68, 73)( 69, 74)( 70, 75)(101,116)(102,117)(103,118)
(104,119)(105,120)(106,111)(107,112)(108,113)(109,114)(110,115)(141,156)
(142,157)(143,158)(144,159)(145,160)(146,151)(147,152)(148,153)(149,154)
(150,155);
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)!(  1,121)(  2,125)(  3,124)(  4,123)(  5,122)(  6,126)(  7,130)
(  8,129)(  9,128)( 10,127)( 11,131)( 12,135)( 13,134)( 14,133)( 15,132)
( 16,136)( 17,140)( 18,139)( 19,138)( 20,137)( 21,141)( 22,145)( 23,144)
( 24,143)( 25,142)( 26,146)( 27,150)( 28,149)( 29,148)( 30,147)( 31,151)
( 32,155)( 33,154)( 34,153)( 35,152)( 36,156)( 37,160)( 38,159)( 39,158)
( 40,157)( 41, 81)( 42, 85)( 43, 84)( 44, 83)( 45, 82)( 46, 86)( 47, 90)
( 48, 89)( 49, 88)( 50, 87)( 51, 91)( 52, 95)( 53, 94)( 54, 93)( 55, 92)
( 56, 96)( 57,100)( 58, 99)( 59, 98)( 60, 97)( 61,101)( 62,105)( 63,104)
( 64,103)( 65,102)( 66,106)( 67,110)( 68,109)( 69,108)( 70,107)( 71,111)
( 72,115)( 73,114)( 74,113)( 75,112)( 76,116)( 77,120)( 78,119)( 79,118)
( 80,117);
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, 
s0*s1*s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

References : None.
to this polytope