Polytope of Type {3,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,6}*648b
if this polytope has a name.
Group : SmallGroup(648,299)
Rank : 4
Schlafli Type : {3,6,6}
Number of vertices, edges, etc : 3, 27, 54, 18
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 6
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,6,6,2} of size 1296
Vertex Figure Of :
{2,3,6,6} of size 1296
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6,3}*324a
3-fold quotients : {3,6,6}*216a
6-fold quotients : {3,6,3}*108
9-fold quotients : {3,2,6}*72
18-fold quotients : {3,2,3}*36
27-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {3,6,12}*1296a, {6,6,6}*1296b
3-fold covers : {3,6,6}*1944a, {3,6,18}*1944a, {9,6,6}*1944b, {3,6,6}*1944f
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, 55)( 29, 57)( 30, 56)( 31, 61)
( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)( 39, 65)
( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)( 47, 75)
( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)( 83, 84)
( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)
(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)(113,144)
(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)(121,151)
(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)(129,155)
(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)(  8, 33)
(  9, 32)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)( 16, 43)
( 17, 45)( 18, 44)( 19, 53)( 20, 52)( 21, 54)( 22, 50)( 23, 49)( 24, 51)
( 25, 47)( 26, 46)( 27, 48)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 67)
( 65, 69)( 66, 68)( 71, 72)( 73, 80)( 74, 79)( 75, 81)( 76, 77)( 82,109)
( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)( 90,113)
( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,124)( 98,126)
( 99,125)(100,134)(101,133)(102,135)(103,131)(104,130)(105,132)(106,128)
(107,127)(108,129)(137,138)(139,142)(140,144)(141,143)(145,148)(146,150)
(147,149)(152,153)(154,161)(155,160)(156,162)(157,158);;
s2 := (  1, 10)(  2, 11)(  3, 12)(  4, 16)(  5, 17)(  6, 18)(  7, 13)(  8, 14)
(  9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)( 31, 43)
( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)( 51, 54)
( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)( 62, 68)
( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)( 85, 97)
( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)(105,108)
(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)(116,122)
(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)(139,151)
(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)(159,162);;
s3 := (  1, 82)(  2, 83)(  3, 84)(  4, 89)(  5, 90)(  6, 88)(  7, 87)(  8, 85)
(  9, 86)( 10,100)( 11,101)( 12,102)( 13,107)( 14,108)( 15,106)( 16,105)
( 17,103)( 18,104)( 19, 91)( 20, 92)( 21, 93)( 22, 98)( 23, 99)( 24, 97)
( 25, 96)( 26, 94)( 27, 95)( 28,109)( 29,110)( 30,111)( 31,116)( 32,117)
( 33,115)( 34,114)( 35,112)( 36,113)( 37,127)( 38,128)( 39,129)( 40,134)
( 41,135)( 42,133)( 43,132)( 44,130)( 45,131)( 46,118)( 47,119)( 48,120)
( 49,125)( 50,126)( 51,124)( 52,123)( 53,121)( 54,122)( 55,136)( 56,137)
( 57,138)( 58,143)( 59,144)( 60,142)( 61,141)( 62,139)( 63,140)( 64,154)
( 65,155)( 66,156)( 67,161)( 68,162)( 69,160)( 70,159)( 71,157)( 72,158)
( 73,145)( 74,146)( 75,147)( 76,152)( 77,153)( 78,151)( 79,150)( 80,148)
( 81,149);;
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, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
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, 55)( 29, 57)( 30, 56)
( 31, 61)( 32, 63)( 33, 62)( 34, 58)( 35, 60)( 36, 59)( 37, 64)( 38, 66)
( 39, 65)( 40, 70)( 41, 72)( 42, 71)( 43, 67)( 44, 69)( 45, 68)( 46, 73)
( 47, 75)( 48, 74)( 49, 79)( 50, 81)( 51, 80)( 52, 76)( 53, 78)( 54, 77)
( 83, 84)( 85, 88)( 86, 90)( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)
(101,102)(103,106)(104,108)(105,107)(109,136)(110,138)(111,137)(112,142)
(113,144)(114,143)(115,139)(116,141)(117,140)(118,145)(119,147)(120,146)
(121,151)(122,153)(123,152)(124,148)(125,150)(126,149)(127,154)(128,156)
(129,155)(130,160)(131,162)(132,161)(133,157)(134,159)(135,158);
s1 := Sym(162)!(  1, 28)(  2, 30)(  3, 29)(  4, 34)(  5, 36)(  6, 35)(  7, 31)
(  8, 33)(  9, 32)( 10, 40)( 11, 42)( 12, 41)( 13, 37)( 14, 39)( 15, 38)
( 16, 43)( 17, 45)( 18, 44)( 19, 53)( 20, 52)( 21, 54)( 22, 50)( 23, 49)
( 24, 51)( 25, 47)( 26, 46)( 27, 48)( 56, 57)( 58, 61)( 59, 63)( 60, 62)
( 64, 67)( 65, 69)( 66, 68)( 71, 72)( 73, 80)( 74, 79)( 75, 81)( 76, 77)
( 82,109)( 83,111)( 84,110)( 85,115)( 86,117)( 87,116)( 88,112)( 89,114)
( 90,113)( 91,121)( 92,123)( 93,122)( 94,118)( 95,120)( 96,119)( 97,124)
( 98,126)( 99,125)(100,134)(101,133)(102,135)(103,131)(104,130)(105,132)
(106,128)(107,127)(108,129)(137,138)(139,142)(140,144)(141,143)(145,148)
(146,150)(147,149)(152,153)(154,161)(155,160)(156,162)(157,158);
s2 := Sym(162)!(  1, 10)(  2, 11)(  3, 12)(  4, 16)(  5, 17)(  6, 18)(  7, 13)
(  8, 14)(  9, 15)( 22, 25)( 23, 26)( 24, 27)( 28, 37)( 29, 38)( 30, 39)
( 31, 43)( 32, 44)( 33, 45)( 34, 40)( 35, 41)( 36, 42)( 49, 52)( 50, 53)
( 51, 54)( 55, 64)( 56, 65)( 57, 66)( 58, 70)( 59, 71)( 60, 72)( 61, 67)
( 62, 68)( 63, 69)( 76, 79)( 77, 80)( 78, 81)( 82, 91)( 83, 92)( 84, 93)
( 85, 97)( 86, 98)( 87, 99)( 88, 94)( 89, 95)( 90, 96)(103,106)(104,107)
(105,108)(109,118)(110,119)(111,120)(112,124)(113,125)(114,126)(115,121)
(116,122)(117,123)(130,133)(131,134)(132,135)(136,145)(137,146)(138,147)
(139,151)(140,152)(141,153)(142,148)(143,149)(144,150)(157,160)(158,161)
(159,162);
s3 := Sym(162)!(  1, 82)(  2, 83)(  3, 84)(  4, 89)(  5, 90)(  6, 88)(  7, 87)
(  8, 85)(  9, 86)( 10,100)( 11,101)( 12,102)( 13,107)( 14,108)( 15,106)
( 16,105)( 17,103)( 18,104)( 19, 91)( 20, 92)( 21, 93)( 22, 98)( 23, 99)
( 24, 97)( 25, 96)( 26, 94)( 27, 95)( 28,109)( 29,110)( 30,111)( 31,116)
( 32,117)( 33,115)( 34,114)( 35,112)( 36,113)( 37,127)( 38,128)( 39,129)
( 40,134)( 41,135)( 42,133)( 43,132)( 44,130)( 45,131)( 46,118)( 47,119)
( 48,120)( 49,125)( 50,126)( 51,124)( 52,123)( 53,121)( 54,122)( 55,136)
( 56,137)( 57,138)( 58,143)( 59,144)( 60,142)( 61,141)( 62,139)( 63,140)
( 64,154)( 65,155)( 66,156)( 67,161)( 68,162)( 69,160)( 70,159)( 71,157)
( 72,158)( 73,145)( 74,146)( 75,147)( 76,152)( 77,153)( 78,151)( 79,150)
( 80,148)( 81,149);
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,
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;

```
References : None.
to this polytope