Polytope of Type {156}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {156}*312
Also Known As : 156-gon, {156}. if this polytope has another name.
Group : SmallGroup(312,39)
Rank : 2
Schlafli Type : {156}
Number of vertices, edges, etc : 156, 156
Order of s0s1 : 156
Special Properties :
Universal
Spherical
Locally Spherical
Orientable
Self-Dual
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{156,2} of size 624
{156,4} of size 1248
{156,4} of size 1248
{156,4} of size 1248
{156,6} of size 1872
{156,6} of size 1872
{156,6} of size 1872
{156,6} of size 1872
Vertex Figure Of :
{2,156} of size 624
{4,156} of size 1248
{4,156} of size 1248
{4,156} of size 1248
{6,156} of size 1872
{6,156} of size 1872
{6,156} of size 1872
{6,156} of size 1872
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {78}*156
3-fold quotients : {52}*104
4-fold quotients : {39}*78
6-fold quotients : {26}*52
12-fold quotients : {13}*26
13-fold quotients : {12}*24
26-fold quotients : {6}*12
39-fold quotients : {4}*8
52-fold quotients : {3}*6
78-fold quotients : {2}*4
Covers (Minimal Covers in Boldface) :
2-fold covers : {312}*624
3-fold covers : {468}*936
4-fold covers : {624}*1248
5-fold covers : {780}*1560
6-fold covers : {936}*1872
Permutation Representation (GAP) :
s0 := ( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)( 15, 39)
( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)
( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)( 45, 48)
( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)( 59, 73)
( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 79,118)( 80,130)
( 81,129)( 82,128)( 83,127)( 84,126)( 85,125)( 86,124)( 87,123)( 88,122)
( 89,121)( 90,120)( 91,119)( 92,144)( 93,156)( 94,155)( 95,154)( 96,153)
( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)(104,145)
(105,131)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)(112,137)
(113,136)(114,135)(115,134)(116,133)(117,132);;
s1 := ( 1, 93)( 2, 92)( 3,104)( 4,103)( 5,102)( 6,101)( 7,100)( 8, 99)
( 9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 80)( 15, 79)( 16, 91)
( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 85)( 23, 84)( 24, 83)
( 25, 82)( 26, 81)( 27,106)( 28,105)( 29,117)( 30,116)( 31,115)( 32,114)
( 33,113)( 34,112)( 35,111)( 36,110)( 37,109)( 38,108)( 39,107)( 40,132)
( 41,131)( 42,143)( 43,142)( 44,141)( 45,140)( 46,139)( 47,138)( 48,137)
( 49,136)( 50,135)( 51,134)( 52,133)( 53,119)( 54,118)( 55,130)( 56,129)
( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)( 64,121)
( 65,120)( 66,145)( 67,144)( 68,156)( 69,155)( 70,154)( 71,153)( 72,152)
( 73,151)( 74,150)( 75,149)( 76,148)( 77,147)( 78,146);;
poly := Group([s0,s1]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1");;
s0 := F.1;; s1 := F.2;;
rels := [ s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*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(156)!( 2, 13)( 3, 12)( 4, 11)( 5, 10)( 6, 9)( 7, 8)( 14, 27)
( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)
( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 41, 52)( 42, 51)( 43, 50)( 44, 49)
( 45, 48)( 46, 47)( 53, 66)( 54, 78)( 55, 77)( 56, 76)( 57, 75)( 58, 74)
( 59, 73)( 60, 72)( 61, 71)( 62, 70)( 63, 69)( 64, 68)( 65, 67)( 79,118)
( 80,130)( 81,129)( 82,128)( 83,127)( 84,126)( 85,125)( 86,124)( 87,123)
( 88,122)( 89,121)( 90,120)( 91,119)( 92,144)( 93,156)( 94,155)( 95,154)
( 96,153)( 97,152)( 98,151)( 99,150)(100,149)(101,148)(102,147)(103,146)
(104,145)(105,131)(106,143)(107,142)(108,141)(109,140)(110,139)(111,138)
(112,137)(113,136)(114,135)(115,134)(116,133)(117,132);
s1 := Sym(156)!( 1, 93)( 2, 92)( 3,104)( 4,103)( 5,102)( 6,101)( 7,100)
( 8, 99)( 9, 98)( 10, 97)( 11, 96)( 12, 95)( 13, 94)( 14, 80)( 15, 79)
( 16, 91)( 17, 90)( 18, 89)( 19, 88)( 20, 87)( 21, 86)( 22, 85)( 23, 84)
( 24, 83)( 25, 82)( 26, 81)( 27,106)( 28,105)( 29,117)( 30,116)( 31,115)
( 32,114)( 33,113)( 34,112)( 35,111)( 36,110)( 37,109)( 38,108)( 39,107)
( 40,132)( 41,131)( 42,143)( 43,142)( 44,141)( 45,140)( 46,139)( 47,138)
( 48,137)( 49,136)( 50,135)( 51,134)( 52,133)( 53,119)( 54,118)( 55,130)
( 56,129)( 57,128)( 58,127)( 59,126)( 60,125)( 61,124)( 62,123)( 63,122)
( 64,121)( 65,120)( 66,145)( 67,144)( 68,156)( 69,155)( 70,154)( 71,153)
( 72,152)( 73,151)( 74,150)( 75,149)( 76,148)( 77,147)( 78,146);
poly := sub<Sym(156)|s0,s1>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1> := Group< s0,s1 | s0*s0, s1*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
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