Polytope of Type {8,6,2,3}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,6,2,3}*1728
if this polytope has a name.
Group : SmallGroup(1728,46549)
Rank : 5
Schlafli Type : {8,6,2,3}
Number of vertices, edges, etc : 24, 72, 18, 3, 3
Order of s0s1s2s3s4 : 24
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Degenerate
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 : {4,6,2,3}*864
4-fold quotients : {4,6,2,3}*432
9-fold quotients : {8,2,2,3}*192
18-fold quotients : {4,2,2,3}*96
36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)( 20, 29)
( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 37, 46)
( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)( 45, 51)
( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 73, 91)( 74, 92)
( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)( 82,100)
( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)( 90,105)
(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)
(117,141)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)(124,130)
(125,131)(126,132);;
s1 := ( 1, 73)( 2, 76)( 3, 79)( 4, 74)( 5, 77)( 6, 80)( 7, 75)( 8, 78)
( 9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)( 16, 84)
( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)( 24,107)
( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)( 32, 95)
( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)( 40,119)
( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)( 48,115)
( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)( 56,130)
( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)( 64,136)
( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)( 72,144);;
s2 := ( 1, 56)( 2, 55)( 3, 57)( 4, 62)( 5, 61)( 6, 63)( 7, 59)( 8, 58)
( 9, 60)( 10, 65)( 11, 64)( 12, 66)( 13, 71)( 14, 70)( 15, 72)( 16, 68)
( 17, 67)( 18, 69)( 19, 47)( 20, 46)( 21, 48)( 22, 53)( 23, 52)( 24, 54)
( 25, 50)( 26, 49)( 27, 51)( 28, 38)( 29, 37)( 30, 39)( 31, 44)( 32, 43)
( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 73,128)( 74,127)( 75,129)( 76,134)
( 77,133)( 78,135)( 79,131)( 80,130)( 81,132)( 82,137)( 83,136)( 84,138)
( 85,143)( 86,142)( 87,144)( 88,140)( 89,139)( 90,141)( 91,119)( 92,118)
( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)( 99,123)(100,110)
(101,109)(102,111)(103,116)(104,115)(105,117)(106,113)(107,112)(108,114);;
s3 := (146,147);;
s4 := (145,146);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s3*s4*s3*s4*s3*s4, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(147)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)
( 20, 29)( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)
( 37, 46)( 38, 47)( 39, 48)( 40, 52)( 41, 53)( 42, 54)( 43, 49)( 44, 50)
( 45, 51)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)( 73, 91)
( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)( 80, 95)( 81, 96)
( 82,100)( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)( 88,103)( 89,104)
( 90,105)(109,136)(110,137)(111,138)(112,142)(113,143)(114,144)(115,139)
(116,140)(117,141)(118,127)(119,128)(120,129)(121,133)(122,134)(123,135)
(124,130)(125,131)(126,132);
s1 := Sym(147)!( 1, 73)( 2, 76)( 3, 79)( 4, 74)( 5, 77)( 6, 80)( 7, 75)
( 8, 78)( 9, 81)( 10, 82)( 11, 85)( 12, 88)( 13, 83)( 14, 86)( 15, 89)
( 16, 84)( 17, 87)( 18, 90)( 19,100)( 20,103)( 21,106)( 22,101)( 23,104)
( 24,107)( 25,102)( 26,105)( 27,108)( 28, 91)( 29, 94)( 30, 97)( 31, 92)
( 32, 95)( 33, 98)( 34, 93)( 35, 96)( 36, 99)( 37,118)( 38,121)( 39,124)
( 40,119)( 41,122)( 42,125)( 43,120)( 44,123)( 45,126)( 46,109)( 47,112)
( 48,115)( 49,110)( 50,113)( 51,116)( 52,111)( 53,114)( 54,117)( 55,127)
( 56,130)( 57,133)( 58,128)( 59,131)( 60,134)( 61,129)( 62,132)( 63,135)
( 64,136)( 65,139)( 66,142)( 67,137)( 68,140)( 69,143)( 70,138)( 71,141)
( 72,144);
s2 := Sym(147)!( 1, 56)( 2, 55)( 3, 57)( 4, 62)( 5, 61)( 6, 63)( 7, 59)
( 8, 58)( 9, 60)( 10, 65)( 11, 64)( 12, 66)( 13, 71)( 14, 70)( 15, 72)
( 16, 68)( 17, 67)( 18, 69)( 19, 47)( 20, 46)( 21, 48)( 22, 53)( 23, 52)
( 24, 54)( 25, 50)( 26, 49)( 27, 51)( 28, 38)( 29, 37)( 30, 39)( 31, 44)
( 32, 43)( 33, 45)( 34, 41)( 35, 40)( 36, 42)( 73,128)( 74,127)( 75,129)
( 76,134)( 77,133)( 78,135)( 79,131)( 80,130)( 81,132)( 82,137)( 83,136)
( 84,138)( 85,143)( 86,142)( 87,144)( 88,140)( 89,139)( 90,141)( 91,119)
( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)( 99,123)
(100,110)(101,109)(102,111)(103,116)(104,115)(105,117)(106,113)(107,112)
(108,114);
s3 := Sym(147)!(146,147);
s4 := Sym(147)!(145,146);
poly := sub<Sym(147)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
to this polytope