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# Polytope of Type {4,6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,6}*864c
if this polytope has a name.
Group : SmallGroup(864,2511)
Rank : 4
Schlafli Type : {4,6,6}
Number of vertices, edges, etc : 4, 36, 54, 18
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{4,6,6,2} of size 1728
Vertex Figure Of :
{2,4,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,6,3}*432a, {2,6,6}*432a
3-fold quotients : {4,6,6}*288c
4-fold quotients : {2,6,3}*216
6-fold quotients : {4,6,3}*144, {2,6,6}*144b
9-fold quotients : {4,2,6}*96
12-fold quotients : {2,6,3}*72
18-fold quotients : {4,2,3}*48, {2,2,6}*48
27-fold quotients : {4,2,2}*32
36-fold quotients : {2,2,3}*24
54-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
2-fold covers : {4,6,12}*1728b, {8,6,6}*1728c, {4,12,6}*1728c
Permutation Representation (GAP) :
```s0 := (  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)(  8, 62)
(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)
( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)
( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)
( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)
( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)
( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)
(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)
(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)
(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)
(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)
(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)
(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)
(159,213)(160,214)(161,215)(162,216);;
s1 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)( 62, 87)
( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)( 70, 94)
( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)( 78,107)
( 79,103)( 80,105)( 81,104)(110,111)(112,115)(113,117)(114,116)(119,120)
(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)
(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)
(157,160)(158,162)(159,161)(163,190)(164,192)(165,191)(166,196)(167,198)
(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)(175,205)
(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)(183,209)
(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);;
s2 := (  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)( 14, 20)
( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)( 37, 49)
( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)( 45, 54)
( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)( 68, 74)
( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)( 91,103)
( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)( 99,108)
(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)(122,128)
(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)(145,157)
(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)(153,162)
(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)(176,182)
(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)(199,211)
(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)(207,216);;
s3 := (  1,118)(  2,119)(  3,120)(  4,126)(  5,124)(  6,125)(  7,122)(  8,123)
(  9,121)( 10,109)( 11,110)( 12,111)( 13,117)( 14,115)( 15,116)( 16,113)
( 17,114)( 18,112)( 19,127)( 20,128)( 21,129)( 22,135)( 23,133)( 24,134)
( 25,131)( 26,132)( 27,130)( 28,145)( 29,146)( 30,147)( 31,153)( 32,151)
( 33,152)( 34,149)( 35,150)( 36,148)( 37,136)( 38,137)( 39,138)( 40,144)
( 41,142)( 42,143)( 43,140)( 44,141)( 45,139)( 46,154)( 47,155)( 48,156)
( 49,162)( 50,160)( 51,161)( 52,158)( 53,159)( 54,157)( 55,172)( 56,173)
( 57,174)( 58,180)( 59,178)( 60,179)( 61,176)( 62,177)( 63,175)( 64,163)
( 65,164)( 66,165)( 67,171)( 68,169)( 69,170)( 70,167)( 71,168)( 72,166)
( 73,181)( 74,182)( 75,183)( 76,189)( 77,187)( 78,188)( 79,185)( 80,186)
( 81,184)( 82,199)( 83,200)( 84,201)( 85,207)( 86,205)( 87,206)( 88,203)
( 89,204)( 90,202)( 91,190)( 92,191)( 93,192)( 94,198)( 95,196)( 96,197)
( 97,194)( 98,195)( 99,193)(100,208)(101,209)(102,210)(103,216)(104,214)
(105,215)(106,212)(107,213)(108,211);;
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*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(216)!(  1, 55)(  2, 56)(  3, 57)(  4, 58)(  5, 59)(  6, 60)(  7, 61)
(  8, 62)(  9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)
( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)
( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)
( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)
( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)
( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)
(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)
(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)
(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)
(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)
(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)
(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)
(158,212)(159,213)(160,214)(161,215)(162,216);
s1 := Sym(216)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 55, 82)( 56, 84)( 57, 83)( 58, 88)( 59, 90)( 60, 89)( 61, 85)
( 62, 87)( 63, 86)( 64, 91)( 65, 93)( 66, 92)( 67, 97)( 68, 99)( 69, 98)
( 70, 94)( 71, 96)( 72, 95)( 73,100)( 74,102)( 75,101)( 76,106)( 77,108)
( 78,107)( 79,103)( 80,105)( 81,104)(110,111)(112,115)(113,117)(114,116)
(119,120)(121,124)(122,126)(123,125)(128,129)(130,133)(131,135)(132,134)
(137,138)(139,142)(140,144)(141,143)(146,147)(148,151)(149,153)(150,152)
(155,156)(157,160)(158,162)(159,161)(163,190)(164,192)(165,191)(166,196)
(167,198)(168,197)(169,193)(170,195)(171,194)(172,199)(173,201)(174,200)
(175,205)(176,207)(177,206)(178,202)(179,204)(180,203)(181,208)(182,210)
(183,209)(184,214)(185,216)(186,215)(187,211)(188,213)(189,212);
s2 := Sym(216)!(  1,  4)(  2,  5)(  3,  6)( 10, 22)( 11, 23)( 12, 24)( 13, 19)
( 14, 20)( 15, 21)( 16, 25)( 17, 26)( 18, 27)( 28, 31)( 29, 32)( 30, 33)
( 37, 49)( 38, 50)( 39, 51)( 40, 46)( 41, 47)( 42, 48)( 43, 52)( 44, 53)
( 45, 54)( 55, 58)( 56, 59)( 57, 60)( 64, 76)( 65, 77)( 66, 78)( 67, 73)
( 68, 74)( 69, 75)( 70, 79)( 71, 80)( 72, 81)( 82, 85)( 83, 86)( 84, 87)
( 91,103)( 92,104)( 93,105)( 94,100)( 95,101)( 96,102)( 97,106)( 98,107)
( 99,108)(109,112)(110,113)(111,114)(118,130)(119,131)(120,132)(121,127)
(122,128)(123,129)(124,133)(125,134)(126,135)(136,139)(137,140)(138,141)
(145,157)(146,158)(147,159)(148,154)(149,155)(150,156)(151,160)(152,161)
(153,162)(163,166)(164,167)(165,168)(172,184)(173,185)(174,186)(175,181)
(176,182)(177,183)(178,187)(179,188)(180,189)(190,193)(191,194)(192,195)
(199,211)(200,212)(201,213)(202,208)(203,209)(204,210)(205,214)(206,215)
(207,216);
s3 := Sym(216)!(  1,118)(  2,119)(  3,120)(  4,126)(  5,124)(  6,125)(  7,122)
(  8,123)(  9,121)( 10,109)( 11,110)( 12,111)( 13,117)( 14,115)( 15,116)
( 16,113)( 17,114)( 18,112)( 19,127)( 20,128)( 21,129)( 22,135)( 23,133)
( 24,134)( 25,131)( 26,132)( 27,130)( 28,145)( 29,146)( 30,147)( 31,153)
( 32,151)( 33,152)( 34,149)( 35,150)( 36,148)( 37,136)( 38,137)( 39,138)
( 40,144)( 41,142)( 42,143)( 43,140)( 44,141)( 45,139)( 46,154)( 47,155)
( 48,156)( 49,162)( 50,160)( 51,161)( 52,158)( 53,159)( 54,157)( 55,172)
( 56,173)( 57,174)( 58,180)( 59,178)( 60,179)( 61,176)( 62,177)( 63,175)
( 64,163)( 65,164)( 66,165)( 67,171)( 68,169)( 69,170)( 70,167)( 71,168)
( 72,166)( 73,181)( 74,182)( 75,183)( 76,189)( 77,187)( 78,188)( 79,185)
( 80,186)( 81,184)( 82,199)( 83,200)( 84,201)( 85,207)( 86,205)( 87,206)
( 88,203)( 89,204)( 90,202)( 91,190)( 92,191)( 93,192)( 94,198)( 95,196)
( 96,197)( 97,194)( 98,195)( 99,193)(100,208)(101,209)(102,210)(103,216)
(104,214)(105,215)(106,212)(107,213)(108,211);
poly := sub<Sym(216)|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*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2*s3*s1*s2*s1*s3*s2 >;

```
References : None.
to this polytope