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# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*648a
if this polytope has a name.
Group : SmallGroup(648,297)
Rank : 3
Schlafli Type : {18,6}
Number of vertices, edges, etc : 54, 162, 18
Order of s0s1s2 : 18
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,6,2} of size 1296
{18,6,3} of size 1944
Vertex Figure Of :
{2,18,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {9,6}*324a
3-fold quotients : {18,6}*216b, {6,6}*216c
6-fold quotients : {9,6}*108, {3,6}*108
9-fold quotients : {18,2}*72, {6,6}*72c
18-fold quotients : {9,2}*36, {3,6}*36
27-fold quotients : {6,2}*24
54-fold quotients : {3,2}*12
81-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {36,6}*1296a, {18,12}*1296e
3-fold covers : {18,18}*1944a, {54,6}*1944a, {18,6}*1944h, {18,18}*1944u, {18,18}*1944y, {18,6}*1944i, {54,6}*1944c, {54,6}*1944e, {18,6}*1944m
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)( 14, 26)
( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 28, 61)( 29, 62)( 30, 63)( 31, 58)
( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 79)( 38, 80)( 39, 81)
( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)( 47, 71)
( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)( 85, 88)
( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)( 96,108)
( 97,103)( 98,104)( 99,105)(109,142)(110,143)(111,144)(112,139)(113,140)
(114,141)(115,136)(116,137)(117,138)(118,160)(119,161)(120,162)(121,157)
(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)(129,153)
(130,148)(131,149)(132,150)(133,145)(134,146)(135,147);;
s1 := (  1,146)(  2,147)(  3,145)(  4,152)(  5,153)(  6,151)(  7,149)(  8,150)
(  9,148)( 10,138)( 11,136)( 12,137)( 13,144)( 14,142)( 15,143)( 16,141)
( 17,139)( 18,140)( 19,154)( 20,155)( 21,156)( 22,160)( 23,161)( 24,162)
( 25,157)( 26,158)( 27,159)( 28,119)( 29,120)( 30,118)( 31,125)( 32,126)
( 33,124)( 34,122)( 35,123)( 36,121)( 37,111)( 38,109)( 39,110)( 40,117)
( 41,115)( 42,116)( 43,114)( 44,112)( 45,113)( 46,127)( 47,128)( 48,129)
( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55, 92)( 56, 93)
( 57, 91)( 58, 98)( 59, 99)( 60, 97)( 61, 95)( 62, 96)( 63, 94)( 64, 84)
( 65, 82)( 66, 83)( 67, 90)( 68, 88)( 69, 89)( 70, 87)( 71, 85)( 72, 86)
( 73,100)( 74,101)( 75,102)( 76,106)( 77,107)( 78,108)( 79,103)( 80,104)
( 81,105);;
s2 := (  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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(162)!(  4,  7)(  5,  8)(  6,  9)( 10, 19)( 11, 20)( 12, 21)( 13, 25)
( 14, 26)( 15, 27)( 16, 22)( 17, 23)( 18, 24)( 28, 61)( 29, 62)( 30, 63)
( 31, 58)( 32, 59)( 33, 60)( 34, 55)( 35, 56)( 36, 57)( 37, 79)( 38, 80)
( 39, 81)( 40, 76)( 41, 77)( 42, 78)( 43, 73)( 44, 74)( 45, 75)( 46, 70)
( 47, 71)( 48, 72)( 49, 67)( 50, 68)( 51, 69)( 52, 64)( 53, 65)( 54, 66)
( 85, 88)( 86, 89)( 87, 90)( 91,100)( 92,101)( 93,102)( 94,106)( 95,107)
( 96,108)( 97,103)( 98,104)( 99,105)(109,142)(110,143)(111,144)(112,139)
(113,140)(114,141)(115,136)(116,137)(117,138)(118,160)(119,161)(120,162)
(121,157)(122,158)(123,159)(124,154)(125,155)(126,156)(127,151)(128,152)
(129,153)(130,148)(131,149)(132,150)(133,145)(134,146)(135,147);
s1 := Sym(162)!(  1,146)(  2,147)(  3,145)(  4,152)(  5,153)(  6,151)(  7,149)
(  8,150)(  9,148)( 10,138)( 11,136)( 12,137)( 13,144)( 14,142)( 15,143)
( 16,141)( 17,139)( 18,140)( 19,154)( 20,155)( 21,156)( 22,160)( 23,161)
( 24,162)( 25,157)( 26,158)( 27,159)( 28,119)( 29,120)( 30,118)( 31,125)
( 32,126)( 33,124)( 34,122)( 35,123)( 36,121)( 37,111)( 38,109)( 39,110)
( 40,117)( 41,115)( 42,116)( 43,114)( 44,112)( 45,113)( 46,127)( 47,128)
( 48,129)( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55, 92)
( 56, 93)( 57, 91)( 58, 98)( 59, 99)( 60, 97)( 61, 95)( 62, 96)( 63, 94)
( 64, 84)( 65, 82)( 66, 83)( 67, 90)( 68, 88)( 69, 89)( 70, 87)( 71, 85)
( 72, 86)( 73,100)( 74,101)( 75,102)( 76,106)( 77,107)( 78,108)( 79,103)
( 80,104)( 81,105);
s2 := Sym(162)!(  2,  3)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)
( 14, 24)( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)
( 37, 46)( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)
( 45, 53)( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)
( 68, 78)( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)
( 91,100)( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)
( 99,107)(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)
(122,132)(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)
(145,154)(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)
(153,161);
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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.
to this polytope