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# Polytope of Type {6,18}

Atlas Canonical Name : {6,18}*648e
if this polytope has a name.
Group : SmallGroup(648,300)
Rank : 3
Schlafli Type : {6,18}
Number of vertices, edges, etc : 18, 162, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
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 : {6,9}*324d
3-fold quotients : {6,6}*216a
6-fold quotients : {6,3}*108
9-fold quotients : {6,6}*72b
18-fold quotients : {6,3}*36
27-fold quotients : {2,6}*24
54-fold quotients : {2,3}*12
81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,36}*1296e, {12,18}*1296h
3-fold covers : {6,18}*1944d, {18,18}*1944j, {18,18}*1944n, {18,18}*1944r, {6,18}*1944i, {6,18}*1944r
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, 65)( 29, 64)( 30, 66)( 31, 68)
( 32, 67)( 33, 69)( 34, 71)( 35, 70)( 36, 72)( 37, 56)( 38, 55)( 39, 57)
( 40, 59)( 41, 58)( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 74)( 47, 73)
( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)( 82, 91)
( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)( 90, 98)
(101,102)(104,105)(107,108)(109,146)(110,145)(111,147)(112,149)(113,148)
(114,150)(115,152)(116,151)(117,153)(118,137)(119,136)(120,138)(121,140)
(122,139)(123,141)(124,143)(125,142)(126,144)(127,155)(128,154)(129,156)
(130,158)(131,157)(132,159)(133,161)(134,160)(135,162);;
s2 := (  1,109)(  2,111)(  3,110)(  4,114)(  5,113)(  6,112)(  7,116)(  8,115)
(  9,117)( 10,134)( 11,133)( 12,135)( 13,127)( 14,129)( 15,128)( 16,132)
( 17,131)( 18,130)( 19,121)( 20,123)( 21,122)( 22,126)( 23,125)( 24,124)
( 25,119)( 26,118)( 27,120)( 28, 82)( 29, 84)( 30, 83)( 31, 87)( 32, 86)
( 33, 85)( 34, 89)( 35, 88)( 36, 90)( 37,107)( 38,106)( 39,108)( 40,100)
( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46, 94)( 47, 96)( 48, 95)
( 49, 99)( 50, 98)( 51, 97)( 52, 92)( 53, 91)( 54, 93)( 55,137)( 56,136)
( 57,138)( 58,139)( 59,141)( 60,140)( 61,144)( 62,143)( 63,142)( 64,162)
( 65,161)( 66,160)( 67,155)( 68,154)( 69,156)( 70,157)( 71,159)( 72,158)
( 73,149)( 74,148)( 75,150)( 76,151)( 77,153)( 78,152)( 79,147)( 80,146)
( 81,145);;
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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2 ];;
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, 65)( 29, 64)( 30, 66)
( 31, 68)( 32, 67)( 33, 69)( 34, 71)( 35, 70)( 36, 72)( 37, 56)( 38, 55)
( 39, 57)( 40, 59)( 41, 58)( 42, 60)( 43, 62)( 44, 61)( 45, 63)( 46, 74)
( 47, 73)( 48, 75)( 49, 77)( 50, 76)( 51, 78)( 52, 80)( 53, 79)( 54, 81)
( 82, 91)( 83, 93)( 84, 92)( 85, 94)( 86, 96)( 87, 95)( 88, 97)( 89, 99)
( 90, 98)(101,102)(104,105)(107,108)(109,146)(110,145)(111,147)(112,149)
(113,148)(114,150)(115,152)(116,151)(117,153)(118,137)(119,136)(120,138)
(121,140)(122,139)(123,141)(124,143)(125,142)(126,144)(127,155)(128,154)
(129,156)(130,158)(131,157)(132,159)(133,161)(134,160)(135,162);
s2 := Sym(162)!(  1,109)(  2,111)(  3,110)(  4,114)(  5,113)(  6,112)(  7,116)
(  8,115)(  9,117)( 10,134)( 11,133)( 12,135)( 13,127)( 14,129)( 15,128)
( 16,132)( 17,131)( 18,130)( 19,121)( 20,123)( 21,122)( 22,126)( 23,125)
( 24,124)( 25,119)( 26,118)( 27,120)( 28, 82)( 29, 84)( 30, 83)( 31, 87)
( 32, 86)( 33, 85)( 34, 89)( 35, 88)( 36, 90)( 37,107)( 38,106)( 39,108)
( 40,100)( 41,102)( 42,101)( 43,105)( 44,104)( 45,103)( 46, 94)( 47, 96)
( 48, 95)( 49, 99)( 50, 98)( 51, 97)( 52, 92)( 53, 91)( 54, 93)( 55,137)
( 56,136)( 57,138)( 58,139)( 59,141)( 60,140)( 61,144)( 62,143)( 63,142)
( 64,162)( 65,161)( 66,160)( 67,155)( 68,154)( 69,156)( 70,157)( 71,159)
( 72,158)( 73,149)( 74,148)( 75,150)( 76,151)( 77,153)( 78,152)( 79,147)
( 80,146)( 81,145);
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,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s2*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2 >;

References : None.
to this polytope