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# Polytope of Type {6,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {6,6,12}
Number of vertices, edges, etc : 6, 36, 72, 24
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) :
2-fold quotients : {6,6,12}*864h
3-fold quotients : {6,6,4}*576b, {2,6,12}*576a
4-fold quotients : {6,6,6}*432d
6-fold quotients : {6,3,4}*288, {2,6,12}*288d, {6,6,4}*288e, {6,6,4}*288f
9-fold quotients : {2,6,4}*192
12-fold quotients : {6,3,4}*144, {2,6,6}*144a, {6,6,2}*144b
18-fold quotients : {2,3,4}*96, {2,6,4}*96b, {2,6,4}*96c
24-fold quotients : {6,3,2}*72
36-fold quotients : {2,3,4}*48, {2,2,6}*48, {2,6,2}*48
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := ( 13, 25)( 14, 26)( 15, 27)( 16, 28)( 17, 29)( 18, 30)( 19, 31)( 20, 32)
( 21, 33)( 22, 34)( 23, 35)( 24, 36)( 49, 61)( 50, 62)( 51, 63)( 52, 64)
( 53, 65)( 54, 66)( 55, 67)( 56, 68)( 57, 69)( 58, 70)( 59, 71)( 60, 72)
( 85, 97)( 86, 98)( 87, 99)( 88,100)( 89,101)( 90,102)( 91,103)( 92,104)
( 93,105)( 94,106)( 95,107)( 96,108)(121,133)(122,134)(123,135)(124,136)
(125,137)(126,138)(127,139)(128,140)(129,141)(130,142)(131,143)(132,144)
(157,169)(158,170)(159,171)(160,172)(161,173)(162,174)(163,175)(164,176)
(165,177)(166,178)(167,179)(168,180)(193,205)(194,206)(195,207)(196,208)
(197,209)(198,210)(199,211)(200,212)(201,213)(202,214)(203,215)(204,216);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 17)(  6, 18)(  7, 20)(  8, 19)
(  9, 21)( 10, 22)( 11, 24)( 12, 23)( 27, 28)( 31, 32)( 35, 36)( 37, 85)
( 38, 86)( 39, 88)( 40, 87)( 41, 89)( 42, 90)( 43, 92)( 44, 91)( 45, 93)
( 46, 94)( 47, 96)( 48, 95)( 49, 73)( 50, 74)( 51, 76)( 52, 75)( 53, 77)
( 54, 78)( 55, 80)( 56, 79)( 57, 81)( 58, 82)( 59, 84)( 60, 83)( 61, 97)
( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)
( 70,106)( 71,108)( 72,107)(109,121)(110,122)(111,124)(112,123)(113,125)
(114,126)(115,128)(116,127)(117,129)(118,130)(119,132)(120,131)(135,136)
(139,140)(143,144)(145,193)(146,194)(147,196)(148,195)(149,197)(150,198)
(151,200)(152,199)(153,201)(154,202)(155,204)(156,203)(157,181)(158,182)
(159,184)(160,183)(161,185)(162,186)(163,188)(164,187)(165,189)(166,190)
(167,192)(168,191)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)
(175,212)(176,211)(177,213)(178,214)(179,216)(180,215);;
s2 := (  1, 37)(  2, 40)(  3, 39)(  4, 38)(  5, 45)(  6, 48)(  7, 47)(  8, 46)
(  9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 61)( 14, 64)( 15, 63)( 16, 62)
( 17, 69)( 18, 72)( 19, 71)( 20, 70)( 21, 65)( 22, 68)( 23, 67)( 24, 66)
( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 57)( 30, 60)( 31, 59)( 32, 58)
( 33, 53)( 34, 56)( 35, 55)( 36, 54)( 74, 76)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,105)( 90,108)( 91,107)
( 92,106)( 93,101)( 94,104)( 95,103)( 96,102)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,169)(122,172)(123,171)(124,170)(125,177)(126,180)(127,179)
(128,178)(129,173)(130,176)(131,175)(132,174)(133,157)(134,160)(135,159)
(136,158)(137,165)(138,168)(139,167)(140,166)(141,161)(142,164)(143,163)
(144,162)(182,184)(185,189)(186,192)(187,191)(188,190)(193,205)(194,208)
(195,207)(196,206)(197,213)(198,216)(199,215)(200,214)(201,209)(202,212)
(203,211)(204,210);;
s3 := (  1,114)(  2,113)(  3,116)(  4,115)(  5,110)(  6,109)(  7,112)(  8,111)
(  9,118)( 10,117)( 11,120)( 12,119)( 13,126)( 14,125)( 15,128)( 16,127)
( 17,122)( 18,121)( 19,124)( 20,123)( 21,130)( 22,129)( 23,132)( 24,131)
( 25,138)( 26,137)( 27,140)( 28,139)( 29,134)( 30,133)( 31,136)( 32,135)
( 33,142)( 34,141)( 35,144)( 36,143)( 37,150)( 38,149)( 39,152)( 40,151)
( 41,146)( 42,145)( 43,148)( 44,147)( 45,154)( 46,153)( 47,156)( 48,155)
( 49,162)( 50,161)( 51,164)( 52,163)( 53,158)( 54,157)( 55,160)( 56,159)
( 57,166)( 58,165)( 59,168)( 60,167)( 61,174)( 62,173)( 63,176)( 64,175)
( 65,170)( 66,169)( 67,172)( 68,171)( 69,178)( 70,177)( 71,180)( 72,179)
( 73,186)( 74,185)( 75,188)( 76,187)( 77,182)( 78,181)( 79,184)( 80,183)
( 81,190)( 82,189)( 83,192)( 84,191)( 85,198)( 86,197)( 87,200)( 88,199)
( 89,194)( 90,193)( 91,196)( 92,195)( 93,202)( 94,201)( 95,204)( 96,203)
( 97,210)( 98,209)( 99,212)(100,211)(101,206)(102,205)(103,208)(104,207)
(105,214)(106,213)(107,216)(108,215);;
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*s0*s1*s2*s0*s1*s0*s1,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s1*s2*s1*s2*s3*s1*s2*s3*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s1*s2*s3*s2*s3*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope