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# Polytope of Type {6,12,6}

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

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

```
References : None.
to this polytope