# Polytope of Type {18,6}

Atlas Canonical Name : {18,6}*648f
if this polytope has a name.
Group : SmallGroup(648,300)
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
Vertex Figure Of :
{2,18,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {18,6}*324b
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 : {18,12}*1296b, {36,6}*1296f
3-fold covers : {18,6}*1944c, {18,18}*1944k, {18,18}*1944m, {18,18}*1944v, {18,6}*1944j, {18,6}*1944s
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 56)( 29, 55)( 30, 57)( 31, 62)
( 32, 61)( 33, 63)( 34, 59)( 35, 58)( 36, 60)( 37, 65)( 38, 64)( 39, 66)
( 40, 71)( 41, 70)( 42, 72)( 43, 68)( 44, 67)( 45, 69)( 46, 74)( 47, 73)
( 48, 75)( 49, 80)( 50, 79)( 51, 81)( 52, 77)( 53, 76)( 54, 78)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,137)(110,136)(111,138)(112,143)(113,142)
(114,144)(115,140)(116,139)(117,141)(118,146)(119,145)(120,147)(121,152)
(122,151)(123,153)(124,149)(125,148)(126,150)(127,155)(128,154)(129,156)
(130,161)(131,160)(132,162)(133,158)(134,157)(135,159);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 53)( 11, 52)( 12, 54)( 13, 46)( 14, 48)( 15, 47)( 16, 51)
( 17, 50)( 18, 49)( 19, 40)( 20, 42)( 21, 41)( 22, 45)( 23, 44)( 24, 43)
( 25, 38)( 26, 37)( 27, 39)( 55, 56)( 59, 60)( 61, 63)( 64, 81)( 65, 80)
( 66, 79)( 67, 74)( 68, 73)( 69, 75)( 70, 76)( 71, 78)( 72, 77)( 82,109)
( 83,111)( 84,110)( 85,114)( 86,113)( 87,112)( 88,116)( 89,115)( 90,117)
( 91,134)( 92,133)( 93,135)( 94,127)( 95,129)( 96,128)( 97,132)( 98,131)
( 99,130)(100,121)(101,123)(102,122)(103,126)(104,125)(105,124)(106,119)
(107,118)(108,120)(136,137)(140,141)(142,144)(145,162)(146,161)(147,160)
(148,155)(149,154)(150,156)(151,157)(152,159)(153,158);;
s2 := (  1, 91)(  2, 92)(  3, 93)(  4, 97)(  5, 98)(  6, 99)(  7, 94)(  8, 95)
(  9, 96)( 10, 82)( 11, 83)( 12, 84)( 13, 88)( 14, 89)( 15, 90)( 16, 85)
( 17, 86)( 18, 87)( 19,100)( 20,101)( 21,102)( 22,106)( 23,107)( 24,108)
( 25,103)( 26,104)( 27,105)( 28,118)( 29,119)( 30,120)( 31,124)( 32,125)
( 33,126)( 34,121)( 35,122)( 36,123)( 37,109)( 38,110)( 39,111)( 40,115)
( 41,116)( 42,117)( 43,112)( 44,113)( 45,114)( 46,127)( 47,128)( 48,129)
( 49,133)( 50,134)( 51,135)( 52,130)( 53,131)( 54,132)( 55,145)( 56,146)
( 57,147)( 58,151)( 59,152)( 60,153)( 61,148)( 62,149)( 63,150)( 64,136)
( 65,137)( 66,138)( 67,142)( 68,143)( 69,144)( 70,139)( 71,140)( 72,141)
( 73,154)( 74,155)( 75,156)( 76,160)( 77,161)( 78,162)( 79,157)( 80,158)
( 81,159);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope