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# Polytope of Type {18,18}

Atlas Canonical Name : {18,18}*648a
Also Known As : {18,18|2}. if this polytope has another 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 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Dual
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) :
3-fold quotients : {6,18}*216a, {18,6}*216a
9-fold quotients : {2,18}*72, {18,2}*72, {6,6}*72a
18-fold quotients : {2,9}*36, {9,2}*36
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 : {18,36}*1296a, {36,18}*1296a
3-fold covers : {18,18}*1944c, {18,54}*1944a, {54,18}*1944a, {18,18}*1944ad
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)(  2,110)(  3,111)(  4,115)(  5,116)(  6,117)(  7,112)(  8,113)
(  9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)( 16,121)
( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)( 24,135)
( 25,130)( 26,131)( 27,132)( 28, 82)( 29, 83)( 30, 84)( 31, 88)( 32, 89)
( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37, 91)( 38, 92)( 39, 93)( 40, 97)
( 41, 98)( 42, 99)( 43, 94)( 44, 95)( 45, 96)( 46,100)( 47,101)( 48,102)
( 49,106)( 50,107)( 51,108)( 52,103)( 53,104)( 54,105)( 55,142)( 56,143)
( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)( 64,151)
( 65,152)( 66,153)( 67,148)( 68,149)( 69,150)( 70,145)( 71,146)( 72,147)
( 73,160)( 74,161)( 75,162)( 76,157)( 77,158)( 78,159)( 79,154)( 80,155)
( 81,156);;
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*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,
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)(  2,110)(  3,111)(  4,115)(  5,116)(  6,117)(  7,112)
(  8,113)(  9,114)( 10,118)( 11,119)( 12,120)( 13,124)( 14,125)( 15,126)
( 16,121)( 17,122)( 18,123)( 19,127)( 20,128)( 21,129)( 22,133)( 23,134)
( 24,135)( 25,130)( 26,131)( 27,132)( 28, 82)( 29, 83)( 30, 84)( 31, 88)
( 32, 89)( 33, 90)( 34, 85)( 35, 86)( 36, 87)( 37, 91)( 38, 92)( 39, 93)
( 40, 97)( 41, 98)( 42, 99)( 43, 94)( 44, 95)( 45, 96)( 46,100)( 47,101)
( 48,102)( 49,106)( 50,107)( 51,108)( 52,103)( 53,104)( 54,105)( 55,142)
( 56,143)( 57,144)( 58,139)( 59,140)( 60,141)( 61,136)( 62,137)( 63,138)
( 64,151)( 65,152)( 66,153)( 67,148)( 68,149)( 69,150)( 70,145)( 71,146)
( 72,147)( 73,160)( 74,161)( 75,162)( 76,157)( 77,158)( 78,159)( 79,154)
( 80,155)( 81,156);
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*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,
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