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# Polytope of Type {6,6,6}

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

```
References : None.
to this polytope