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# Polytope of Type {54,6}

Atlas Canonical Name : {54,6}*648b
if this polytope has a name.
Group : SmallGroup(648,298)
Rank : 3
Schlafli Type : {54,6}
Number of vertices, edges, etc : 54, 162, 6
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{54,6,2} of size 1296
{54,6,3} of size 1944
Vertex Figure Of :
{2,54,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {27,6}*324
3-fold quotients : {54,2}*216, {18,6}*216b
6-fold quotients : {27,2}*108, {9,6}*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 : {108,6}*1296b, {54,12}*1296b
3-fold covers : {54,18}*1944b, {54,6}*1944a, {162,6}*1944b, {54,6}*1944g
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 19)( 11, 21)( 12, 20)( 13, 27)
( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 63)( 29, 62)( 30, 61)
( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 81)( 38, 80)
( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)( 46, 72)
( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)( 54, 64)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,100)( 92,102)( 93,101)( 94,108)
( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,144)(110,143)(111,142)
(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,162)(119,161)
(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)(127,153)
(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)(135,145);;
s1 := (  1,118)(  2,120)(  3,119)(  4,126)(  5,125)(  6,124)(  7,123)(  8,122)
(  9,121)( 10,109)( 11,111)( 12,110)( 13,117)( 14,116)( 15,115)( 16,114)
( 17,113)( 18,112)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,131)( 27,130)( 28, 91)( 29, 93)( 30, 92)( 31, 99)( 32, 98)
( 33, 97)( 34, 96)( 35, 95)( 36, 94)( 37, 82)( 38, 84)( 39, 83)( 40, 90)
( 41, 89)( 42, 88)( 43, 87)( 44, 86)( 45, 85)( 46,100)( 47,102)( 48,101)
( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,153)( 56,152)
( 57,151)( 58,150)( 59,149)( 60,148)( 61,147)( 62,146)( 63,145)( 64,144)
( 65,143)( 66,142)( 67,141)( 68,140)( 69,139)( 70,138)( 71,137)( 72,136)
( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)( 80,155)
( 81,154);;
s2 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)(121,130)
(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)(147,156)
(148,157)(149,158)(150,159)(151,160)(152,161)(153,162);;
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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*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*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*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)!(  2,  3)(  4,  9)(  5,  8)(  6,  7)( 10, 19)( 11, 21)( 12, 20)
( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)( 18, 22)( 28, 63)( 29, 62)
( 30, 61)( 31, 60)( 32, 59)( 33, 58)( 34, 57)( 35, 56)( 36, 55)( 37, 81)
( 38, 80)( 39, 79)( 40, 78)( 41, 77)( 42, 76)( 43, 75)( 44, 74)( 45, 73)
( 46, 72)( 47, 71)( 48, 70)( 49, 69)( 50, 68)( 51, 67)( 52, 66)( 53, 65)
( 54, 64)( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,100)( 92,102)( 93,101)
( 94,108)( 95,107)( 96,106)( 97,105)( 98,104)( 99,103)(109,144)(110,143)
(111,142)(112,141)(113,140)(114,139)(115,138)(116,137)(117,136)(118,162)
(119,161)(120,160)(121,159)(122,158)(123,157)(124,156)(125,155)(126,154)
(127,153)(128,152)(129,151)(130,150)(131,149)(132,148)(133,147)(134,146)
(135,145);
s1 := Sym(162)!(  1,118)(  2,120)(  3,119)(  4,126)(  5,125)(  6,124)(  7,123)
(  8,122)(  9,121)( 10,109)( 11,111)( 12,110)( 13,117)( 14,116)( 15,115)
( 16,114)( 17,113)( 18,112)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)
( 24,133)( 25,132)( 26,131)( 27,130)( 28, 91)( 29, 93)( 30, 92)( 31, 99)
( 32, 98)( 33, 97)( 34, 96)( 35, 95)( 36, 94)( 37, 82)( 38, 84)( 39, 83)
( 40, 90)( 41, 89)( 42, 88)( 43, 87)( 44, 86)( 45, 85)( 46,100)( 47,102)
( 48,101)( 49,108)( 50,107)( 51,106)( 52,105)( 53,104)( 54,103)( 55,153)
( 56,152)( 57,151)( 58,150)( 59,149)( 60,148)( 61,147)( 62,146)( 63,145)
( 64,144)( 65,143)( 66,142)( 67,141)( 68,140)( 69,139)( 70,138)( 71,137)
( 72,136)( 73,162)( 74,161)( 75,160)( 76,159)( 77,158)( 78,157)( 79,156)
( 80,155)( 81,154);
s2 := Sym(162)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(118,127)(119,128)(120,129)
(121,130)(122,131)(123,132)(124,133)(125,134)(126,135)(145,154)(146,155)
(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)(153,162);
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*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*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*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*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