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# Polytope of Type {6,6,6}

Atlas Canonical Name : {6,6,6}*1296p
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 6, 54, 54, 18
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,6}*648e
3-fold quotients : {6,6,6}*432a, {6,6,6}*432g
6-fold quotients : {3,6,6}*216a, {6,6,3}*216a, {3,6,6}*216b
9-fold quotients : {6,2,6}*144, {6,6,2}*144c
12-fold quotients : {3,6,3}*108
18-fold quotients : {3,2,6}*72, {3,6,2}*72, {6,2,3}*72
27-fold quotients : {2,2,6}*48, {6,2,2}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,3}*24, {3,2,2}*24
81-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)( 13, 25)
( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)( 43, 49)
( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)( 65, 75)
( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 91,100)( 92,102)( 93,101)( 94,106)( 95,108)
( 96,107)( 97,103)( 98,105)( 99,104)(110,111)(112,115)(113,117)(114,116)
(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)(125,132)
(126,131)(137,138)(139,142)(140,144)(141,143)(145,154)(146,156)(147,155)
(148,160)(149,162)(150,161)(151,157)(152,159)(153,158);;
s1 := (  1, 94)(  2, 96)(  3, 95)(  4, 91)(  5, 93)(  6, 92)(  7, 97)(  8, 99)
(  9, 98)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)( 16, 88)
( 17, 90)( 18, 89)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)( 24,101)
( 25,106)( 26,108)( 27,107)( 28,121)( 29,123)( 30,122)( 31,118)( 32,120)
( 33,119)( 34,124)( 35,126)( 36,125)( 37,112)( 38,114)( 39,113)( 40,109)
( 41,111)( 42,110)( 43,115)( 44,117)( 45,116)( 46,130)( 47,132)( 48,131)
( 49,127)( 50,129)( 51,128)( 52,133)( 53,135)( 54,134)( 55,148)( 56,150)
( 57,149)( 58,145)( 59,147)( 60,146)( 61,151)( 62,153)( 63,152)( 64,139)
( 65,141)( 66,140)( 67,136)( 68,138)( 69,137)( 70,142)( 71,144)( 72,143)
( 73,157)( 74,159)( 75,158)( 76,154)( 77,156)( 78,155)( 79,160)( 80,162)
( 81,161);;
s2 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)( 16, 41)
( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)( 24, 52)
( 25, 51)( 26, 50)( 27, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 65)
( 67, 71)( 68, 70)( 69, 72)( 73, 75)( 76, 81)( 77, 80)( 78, 79)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)( 98,121)
( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)(106,132)
(107,131)(108,130)(137,138)(139,142)(140,144)(141,143)(145,146)(148,152)
(149,151)(150,153)(154,156)(157,162)(158,161)(159,160);;
s3 := (  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)
( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)( 34, 61)
( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)( 42, 68)
( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)( 50, 78)
( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 83, 84)( 86, 87)( 89, 90)( 92, 93)
( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)(111,137)
(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)(119,147)
(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)(127,154)
(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)(135,161);;
poly := Group([s0,s1,s2,s3]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(162)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 19)( 11, 21)( 12, 20)
( 13, 25)( 14, 27)( 15, 26)( 16, 22)( 17, 24)( 18, 23)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 46)( 38, 48)( 39, 47)( 40, 52)( 41, 54)( 42, 53)
( 43, 49)( 44, 51)( 45, 50)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 73)
( 65, 75)( 66, 74)( 67, 79)( 68, 81)( 69, 80)( 70, 76)( 71, 78)( 72, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 91,100)( 92,102)( 93,101)( 94,106)
( 95,108)( 96,107)( 97,103)( 98,105)( 99,104)(110,111)(112,115)(113,117)
(114,116)(118,127)(119,129)(120,128)(121,133)(122,135)(123,134)(124,130)
(125,132)(126,131)(137,138)(139,142)(140,144)(141,143)(145,154)(146,156)
(147,155)(148,160)(149,162)(150,161)(151,157)(152,159)(153,158);
s1 := Sym(162)!(  1, 94)(  2, 96)(  3, 95)(  4, 91)(  5, 93)(  6, 92)(  7, 97)
(  8, 99)(  9, 98)( 10, 85)( 11, 87)( 12, 86)( 13, 82)( 14, 84)( 15, 83)
( 16, 88)( 17, 90)( 18, 89)( 19,103)( 20,105)( 21,104)( 22,100)( 23,102)
( 24,101)( 25,106)( 26,108)( 27,107)( 28,121)( 29,123)( 30,122)( 31,118)
( 32,120)( 33,119)( 34,124)( 35,126)( 36,125)( 37,112)( 38,114)( 39,113)
( 40,109)( 41,111)( 42,110)( 43,115)( 44,117)( 45,116)( 46,130)( 47,132)
( 48,131)( 49,127)( 50,129)( 51,128)( 52,133)( 53,135)( 54,134)( 55,148)
( 56,150)( 57,149)( 58,145)( 59,147)( 60,146)( 61,151)( 62,153)( 63,152)
( 64,139)( 65,141)( 66,140)( 67,136)( 68,138)( 69,137)( 70,142)( 71,144)
( 72,143)( 73,157)( 74,159)( 75,158)( 76,154)( 77,156)( 78,155)( 79,160)
( 80,162)( 81,161);
s2 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 38)( 11, 37)( 12, 39)( 13, 44)( 14, 43)( 15, 45)
( 16, 41)( 17, 40)( 18, 42)( 19, 48)( 20, 47)( 21, 46)( 22, 54)( 23, 53)
( 24, 52)( 25, 51)( 26, 50)( 27, 49)( 56, 57)( 58, 61)( 59, 63)( 60, 62)
( 64, 65)( 67, 71)( 68, 70)( 69, 72)( 73, 75)( 76, 81)( 77, 80)( 78, 79)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,119)( 92,118)( 93,120)( 94,125)( 95,124)( 96,126)( 97,122)
( 98,121)( 99,123)(100,129)(101,128)(102,127)(103,135)(104,134)(105,133)
(106,132)(107,131)(108,130)(137,138)(139,142)(140,144)(141,143)(145,146)
(148,152)(149,151)(150,153)(154,156)(157,162)(158,161)(159,160);
s3 := Sym(162)!(  2,  3)(  5,  6)(  8,  9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)
( 23, 24)( 26, 27)( 28, 55)( 29, 57)( 30, 56)( 31, 58)( 32, 60)( 33, 59)
( 34, 61)( 35, 63)( 36, 62)( 37, 64)( 38, 66)( 39, 65)( 40, 67)( 41, 69)
( 42, 68)( 43, 70)( 44, 72)( 45, 71)( 46, 73)( 47, 75)( 48, 74)( 49, 76)
( 50, 78)( 51, 77)( 52, 79)( 53, 81)( 54, 80)( 83, 84)( 86, 87)( 89, 90)
( 92, 93)( 95, 96)( 98, 99)(101,102)(104,105)(107,108)(109,136)(110,138)
(111,137)(112,139)(113,141)(114,140)(115,142)(116,144)(117,143)(118,145)
(119,147)(120,146)(121,148)(122,150)(123,149)(124,151)(125,153)(126,152)
(127,154)(128,156)(129,155)(130,157)(131,159)(132,158)(133,160)(134,162)
(135,161);
poly := sub<Sym(162)|s0,s1,s2,s3>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2 >;

```
References : None.
to this polytope