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# Polytope of Type {6,6,6}

Atlas Canonical Name : {6,6,6}*1296n
if this polytope has a name.
Group : SmallGroup(1296,2985)
Rank : 4
Schlafli Type : {6,6,6}
Number of vertices, edges, etc : 18, 54, 54, 6
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 : {6,6,3}*648e
3-fold quotients : {6,6,6}*432a, {6,6,6}*432c
6-fold quotients : {3,6,6}*216a, {6,6,3}*216a, {6,6,3}*216b
9-fold quotients : {2,6,6}*144b, {6,2,6}*144
12-fold quotients : {3,6,3}*108
18-fold quotients : {2,6,3}*72, {3,2,6}*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)(  5,  6)(  8,  9)( 10, 19)( 11, 21)( 12, 20)( 13, 22)( 14, 24)
( 15, 23)( 16, 25)( 17, 27)( 18, 26)( 29, 30)( 32, 33)( 35, 36)( 37, 46)
( 38, 48)( 39, 47)( 40, 49)( 41, 51)( 42, 50)( 43, 52)( 44, 54)( 45, 53)
( 56, 57)( 59, 60)( 62, 63)( 64, 73)( 65, 75)( 66, 74)( 67, 76)( 68, 78)
( 69, 77)( 70, 79)( 71, 81)( 72, 80)( 83, 84)( 86, 87)( 89, 90)( 91,100)
( 92,102)( 93,101)( 94,103)( 95,105)( 96,104)( 97,106)( 98,108)( 99,107)
(110,111)(113,114)(116,117)(118,127)(119,129)(120,128)(121,130)(122,132)
(123,131)(124,133)(125,135)(126,134)(137,138)(140,141)(143,144)(145,154)
(146,156)(147,155)(148,157)(149,159)(150,158)(151,160)(152,162)(153,161);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 16)(  5, 18)(  6, 17)(  7, 13)(  8, 15)
(  9, 14)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 28, 37)( 29, 39)( 30, 38)
( 31, 43)( 32, 45)( 33, 44)( 34, 40)( 35, 42)( 36, 41)( 47, 48)( 49, 52)
( 50, 54)( 51, 53)( 55, 64)( 56, 66)( 57, 65)( 58, 70)( 59, 72)( 60, 71)
( 61, 67)( 62, 69)( 63, 68)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 82, 91)
( 83, 93)( 84, 92)( 85, 97)( 86, 99)( 87, 98)( 88, 94)( 89, 96)( 90, 95)
(101,102)(103,106)(104,108)(105,107)(109,118)(110,120)(111,119)(112,124)
(113,126)(114,125)(115,121)(116,123)(117,122)(128,129)(130,133)(131,135)
(132,134)(136,145)(137,147)(138,146)(139,151)(140,153)(141,152)(142,148)
(143,150)(144,149)(155,156)(157,160)(158,162)(159,161);;
s2 := (  1, 31)(  2, 33)(  3, 32)(  4, 28)(  5, 30)(  6, 29)(  7, 34)(  8, 36)
(  9, 35)( 10, 41)( 11, 40)( 12, 42)( 13, 38)( 14, 37)( 15, 39)( 16, 44)
( 17, 43)( 18, 45)( 19, 51)( 20, 50)( 21, 49)( 22, 48)( 23, 47)( 24, 46)
( 25, 54)( 26, 53)( 27, 52)( 55, 58)( 56, 60)( 57, 59)( 62, 63)( 64, 68)
( 65, 67)( 66, 69)( 70, 71)( 73, 78)( 74, 77)( 75, 76)( 79, 81)( 82,112)
( 83,114)( 84,113)( 85,109)( 86,111)( 87,110)( 88,115)( 89,117)( 90,116)
( 91,122)( 92,121)( 93,123)( 94,119)( 95,118)( 96,120)( 97,125)( 98,124)
( 99,126)(100,132)(101,131)(102,130)(103,129)(104,128)(105,127)(106,135)
(107,134)(108,133)(136,139)(137,141)(138,140)(143,144)(145,149)(146,148)
(147,150)(151,152)(154,159)(155,158)(156,157)(160,162);;
s3 := (  1, 82)(  2, 84)(  3, 83)(  4, 88)(  5, 90)(  6, 89)(  7, 85)(  8, 87)
(  9, 86)( 10, 91)( 11, 93)( 12, 92)( 13, 97)( 14, 99)( 15, 98)( 16, 94)
( 17, 96)( 18, 95)( 19,100)( 20,102)( 21,101)( 22,106)( 23,108)( 24,107)
( 25,103)( 26,105)( 27,104)( 28,136)( 29,138)( 30,137)( 31,142)( 32,144)
( 33,143)( 34,139)( 35,141)( 36,140)( 37,145)( 38,147)( 39,146)( 40,151)
( 41,153)( 42,152)( 43,148)( 44,150)( 45,149)( 46,154)( 47,156)( 48,155)
( 49,160)( 50,162)( 51,161)( 52,157)( 53,159)( 54,158)( 55,109)( 56,111)
( 57,110)( 58,115)( 59,117)( 60,116)( 61,112)( 62,114)( 63,113)( 64,118)
( 65,120)( 66,119)( 67,124)( 68,126)( 69,125)( 70,121)( 71,123)( 72,122)
( 73,127)( 74,129)( 75,128)( 76,133)( 77,135)( 78,134)( 79,130)( 80,132)
( 81,131);;
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, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope