Polytope of Type {6,18}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,18}*648g
if this polytope has a name.
Group : SmallGroup(648,301)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 18, 162, 54
Order of s0s1s2 : 6
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{6,18,2} of size 1296
Vertex Figure Of :
{2,6,18} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,18}*324c
3-fold quotients : {6,6}*216b
6-fold quotients : {6,6}*108
9-fold quotients : {6,6}*72a
27-fold quotients : {2,6}*24, {6,2}*24
54-fold quotients : {2,3}*12, {3,2}*12
81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {12,18}*1296c, {6,36}*1296g
3-fold covers : {18,18}*1944d, {6,18}*1944e, {18,18}*1944k, {18,18}*1944x, {6,18}*1944j, {6,18}*1944l, {6,18}*1944t
Permutation Representation (GAP) :
s0 := ( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)( 14, 27)
( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)( 37, 46)
( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)( 45, 50)
( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)( 68, 81)
( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)( 91,100)
( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)( 99,104)
(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)(122,135)
(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)(145,154)
(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)(153,158);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)( 8, 18)
( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)( 31, 69)
( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)( 39, 55)
( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)( 47, 74)
( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)( 82, 91)
( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)
(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)(113,149)
(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)(121,141)
(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)(129,154)
(130,159)(131,158)(132,157)(133,162)(134,161)(135,160);;
s2 := ( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)( 8,114)
( 9,113)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)( 16,124)
( 17,126)( 18,125)( 19,134)( 20,133)( 21,135)( 22,131)( 23,130)( 24,132)
( 25,128)( 26,127)( 27,129)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)
( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37, 94)( 38, 96)( 39, 95)( 40, 91)
( 41, 93)( 42, 92)( 43, 97)( 44, 99)( 45, 98)( 46,107)( 47,106)( 48,108)
( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102)( 55,138)( 56,137)
( 57,136)( 58,144)( 59,143)( 60,142)( 61,141)( 62,140)( 63,139)( 64,150)
( 65,149)( 66,148)( 67,147)( 68,146)( 69,145)( 70,153)( 71,152)( 72,151)
( 73,160)( 74,162)( 75,161)( 76,157)( 77,159)( 78,158)( 79,154)( 80,156)
( 81,155);;
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*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(162)!( 4, 8)( 5, 9)( 6, 7)( 10, 19)( 11, 20)( 12, 21)( 13, 26)
( 14, 27)( 15, 25)( 16, 24)( 17, 22)( 18, 23)( 31, 35)( 32, 36)( 33, 34)
( 37, 46)( 38, 47)( 39, 48)( 40, 53)( 41, 54)( 42, 52)( 43, 51)( 44, 49)
( 45, 50)( 58, 62)( 59, 63)( 60, 61)( 64, 73)( 65, 74)( 66, 75)( 67, 80)
( 68, 81)( 69, 79)( 70, 78)( 71, 76)( 72, 77)( 85, 89)( 86, 90)( 87, 88)
( 91,100)( 92,101)( 93,102)( 94,107)( 95,108)( 96,106)( 97,105)( 98,103)
( 99,104)(112,116)(113,117)(114,115)(118,127)(119,128)(120,129)(121,134)
(122,135)(123,133)(124,132)(125,130)(126,131)(139,143)(140,144)(141,142)
(145,154)(146,155)(147,156)(148,161)(149,162)(150,160)(151,159)(152,157)
(153,158);
s1 := Sym(162)!( 1, 10)( 2, 12)( 3, 11)( 4, 13)( 5, 15)( 6, 14)( 7, 16)
( 8, 18)( 9, 17)( 20, 21)( 23, 24)( 26, 27)( 28, 66)( 29, 65)( 30, 64)
( 31, 69)( 32, 68)( 33, 67)( 34, 72)( 35, 71)( 36, 70)( 37, 57)( 38, 56)
( 39, 55)( 40, 60)( 41, 59)( 42, 58)( 43, 63)( 44, 62)( 45, 61)( 46, 75)
( 47, 74)( 48, 73)( 49, 78)( 50, 77)( 51, 76)( 52, 81)( 53, 80)( 54, 79)
( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)
( 90, 98)(101,102)(104,105)(107,108)(109,147)(110,146)(111,145)(112,150)
(113,149)(114,148)(115,153)(116,152)(117,151)(118,138)(119,137)(120,136)
(121,141)(122,140)(123,139)(124,144)(125,143)(126,142)(127,156)(128,155)
(129,154)(130,159)(131,158)(132,157)(133,162)(134,161)(135,160);
s2 := Sym(162)!( 1,109)( 2,111)( 3,110)( 4,115)( 5,117)( 6,116)( 7,112)
( 8,114)( 9,113)( 10,121)( 11,123)( 12,122)( 13,118)( 14,120)( 15,119)
( 16,124)( 17,126)( 18,125)( 19,134)( 20,133)( 21,135)( 22,131)( 23,130)
( 24,132)( 25,128)( 26,127)( 27,129)( 28, 82)( 29, 84)( 30, 83)( 31, 88)
( 32, 90)( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37, 94)( 38, 96)( 39, 95)
( 40, 91)( 41, 93)( 42, 92)( 43, 97)( 44, 99)( 45, 98)( 46,107)( 47,106)
( 48,108)( 49,104)( 50,103)( 51,105)( 52,101)( 53,100)( 54,102)( 55,138)
( 56,137)( 57,136)( 58,144)( 59,143)( 60,142)( 61,141)( 62,140)( 63,139)
( 64,150)( 65,149)( 66,148)( 67,147)( 68,146)( 69,145)( 70,153)( 71,152)
( 72,151)( 73,160)( 74,162)( 75,161)( 76,157)( 77,159)( 78,158)( 79,154)
( 80,156)( 81,155);
poly := sub<Sym(162)|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*s0*s1*s0*s1,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References : None.
to this polytope