# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*648b
if this polytope has a name.
Group : SmallGroup(648,296)
Rank : 3
Schlafli Type : {18,18}
Number of vertices, edges, etc : 18, 162, 18
Order of s0s1s2 : 18
Order of s0s1s2s1 : 18
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{18,18,2} of size 1296
Vertex Figure Of :
{2,18,18} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,9}*324
3-fold quotients : {6,18}*216b
6-fold quotients : {6,9}*108
9-fold quotients : {2,18}*72, {6,6}*72b
18-fold quotients : {2,9}*36, {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 : {18,36}*1296b, {36,18}*1296c
3-fold covers : {18,18}*1944b, {18,54}*1944b, {18,18}*1944ae
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  6)(  8,  9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)( 37, 48)
( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)( 45, 52)
( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)( 68, 77)
( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)( 91,102)
( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)( 99,106)
(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)(122,131)
(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)(145,156)
(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)(153,160);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 70)( 29, 72)( 30, 71)
( 31, 67)( 32, 69)( 33, 68)( 34, 64)( 35, 66)( 36, 65)( 37, 61)( 38, 63)
( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)( 46, 81)
( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)( 54, 73)
( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)
( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,151)(110,153)(111,152)
(112,148)(113,150)(114,149)(115,145)(116,147)(117,146)(118,142)(119,144)
(120,143)(121,139)(122,141)(123,140)(124,136)(125,138)(126,137)(127,162)
(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)(135,154);;
s2 := (  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)(  8,114)
(  9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)( 16,132)
( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)( 24,124)
( 25,123)( 26,122)( 27,121)( 28, 82)( 29, 84)( 30, 83)( 31, 88)( 32, 90)
( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37,102)( 38,101)( 39,100)( 40,108)
( 41,107)( 42,106)( 43,105)( 44,104)( 45,103)( 46, 93)( 47, 92)( 48, 91)
( 49, 99)( 50, 98)( 51, 97)( 52, 96)( 53, 95)( 54, 94)( 55,142)( 56,144)
( 57,143)( 58,139)( 59,141)( 60,140)( 61,136)( 62,138)( 63,137)( 64,162)
( 65,161)( 66,160)( 67,159)( 68,158)( 69,157)( 70,156)( 71,155)( 72,154)
( 73,153)( 74,152)( 75,151)( 76,150)( 77,149)( 78,148)( 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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(162)!(  2,  3)(  5,  6)(  8,  9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 29, 30)( 32, 33)( 35, 36)
( 37, 48)( 38, 47)( 39, 46)( 40, 51)( 41, 50)( 42, 49)( 43, 54)( 44, 53)
( 45, 52)( 56, 57)( 59, 60)( 62, 63)( 64, 75)( 65, 74)( 66, 73)( 67, 78)
( 68, 77)( 69, 76)( 70, 81)( 71, 80)( 72, 79)( 83, 84)( 86, 87)( 89, 90)
( 91,102)( 92,101)( 93,100)( 94,105)( 95,104)( 96,103)( 97,108)( 98,107)
( 99,106)(110,111)(113,114)(116,117)(118,129)(119,128)(120,127)(121,132)
(122,131)(123,130)(124,135)(125,134)(126,133)(137,138)(140,141)(143,144)
(145,156)(146,155)(147,154)(148,159)(149,158)(150,157)(151,162)(152,161)
(153,160);
s1 := Sym(162)!(  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)
(  8, 15)(  9, 14)( 19, 21)( 22, 27)( 23, 26)( 24, 25)( 28, 70)( 29, 72)
( 30, 71)( 31, 67)( 32, 69)( 33, 68)( 34, 64)( 35, 66)( 36, 65)( 37, 61)
( 38, 63)( 39, 62)( 40, 58)( 41, 60)( 42, 59)( 43, 55)( 44, 57)( 45, 56)
( 46, 81)( 47, 80)( 48, 79)( 49, 78)( 50, 77)( 51, 76)( 52, 75)( 53, 74)
( 54, 73)( 82, 91)( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)
( 89, 96)( 90, 95)(100,102)(103,108)(104,107)(105,106)(109,151)(110,153)
(111,152)(112,148)(113,150)(114,149)(115,145)(116,147)(117,146)(118,142)
(119,144)(120,143)(121,139)(122,141)(123,140)(124,136)(125,138)(126,137)
(127,162)(128,161)(129,160)(130,159)(131,158)(132,157)(133,156)(134,155)
(135,154);
s2 := Sym(162)!(  1,109)(  2,111)(  3,110)(  4,115)(  5,117)(  6,116)(  7,112)
(  8,114)(  9,113)( 10,129)( 11,128)( 12,127)( 13,135)( 14,134)( 15,133)
( 16,132)( 17,131)( 18,130)( 19,120)( 20,119)( 21,118)( 22,126)( 23,125)
( 24,124)( 25,123)( 26,122)( 27,121)( 28, 82)( 29, 84)( 30, 83)( 31, 88)
( 32, 90)( 33, 89)( 34, 85)( 35, 87)( 36, 86)( 37,102)( 38,101)( 39,100)
( 40,108)( 41,107)( 42,106)( 43,105)( 44,104)( 45,103)( 46, 93)( 47, 92)
( 48, 91)( 49, 99)( 50, 98)( 51, 97)( 52, 96)( 53, 95)( 54, 94)( 55,142)
( 56,144)( 57,143)( 58,139)( 59,141)( 60,140)( 61,136)( 62,138)( 63,137)
( 64,162)( 65,161)( 66,160)( 67,159)( 68,158)( 69,157)( 70,156)( 71,155)
( 72,154)( 73,153)( 74,152)( 75,151)( 76,150)( 77,149)( 78,148)( 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, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```
References : None.
to this polytope