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# Polytope of Type {2,12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,12}*1728a
if this polytope has a name.
Group : SmallGroup(1728,16947)
Rank : 4
Schlafli Type : {2,12,12}
Number of vertices, edges, etc : 2, 36, 216, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Degenerate
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 : {2,12,6}*864a, {2,6,12}*864c
3-fold quotients : {2,12,12}*576c
4-fold quotients : {2,6,6}*432c
6-fold quotients : {2,12,6}*288b, {2,6,12}*288c
8-fold quotients : {2,3,6}*216
9-fold quotients : {2,12,4}*192a
12-fold quotients : {2,6,6}*144c
18-fold quotients : {2,12,2}*96, {2,6,4}*96a
24-fold quotients : {2,3,6}*72
27-fold quotients : {2,4,4}*64
36-fold quotients : {2,6,2}*48
54-fold quotients : {2,2,4}*32, {2,4,2}*32
72-fold quotients : {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (1,2);;
s1 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)( 70, 82)
( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)( 93,102)
( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)(101,107)
(111,192)(112,193)(113,194)(114,198)(115,199)(116,200)(117,195)(118,196)
(119,197)(120,210)(121,211)(122,212)(123,216)(124,217)(125,218)(126,213)
(127,214)(128,215)(129,201)(130,202)(131,203)(132,207)(133,208)(134,209)
(135,204)(136,205)(137,206)(138,165)(139,166)(140,167)(141,171)(142,172)
(143,173)(144,168)(145,169)(146,170)(147,183)(148,184)(149,185)(150,189)
(151,190)(152,191)(153,186)(154,187)(155,188)(156,174)(157,175)(158,176)
(159,180)(160,181)(161,182)(162,177)(163,178)(164,179);;
s2 := (  3,133)(  4,134)(  5,132)(  6,131)(  7,129)(  8,130)(  9,135)( 10,136)
( 11,137)( 12,124)( 13,125)( 14,123)( 15,122)( 16,120)( 17,121)( 18,126)
( 19,127)( 20,128)( 21,115)( 22,116)( 23,114)( 24,113)( 25,111)( 26,112)
( 27,117)( 28,118)( 29,119)( 30,160)( 31,161)( 32,159)( 33,158)( 34,156)
( 35,157)( 36,162)( 37,163)( 38,164)( 39,151)( 40,152)( 41,150)( 42,149)
( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,142)( 49,143)( 50,141)
( 51,140)( 52,138)( 53,139)( 54,144)( 55,145)( 56,146)( 57,187)( 58,188)
( 59,186)( 60,185)( 61,183)( 62,184)( 63,189)( 64,190)( 65,191)( 66,178)
( 67,179)( 68,177)( 69,176)( 70,174)( 71,175)( 72,180)( 73,181)( 74,182)
( 75,169)( 76,170)( 77,168)( 78,167)( 79,165)( 80,166)( 81,171)( 82,172)
( 83,173)( 84,214)( 85,215)( 86,213)( 87,212)( 88,210)( 89,211)( 90,216)
( 91,217)( 92,218)( 93,205)( 94,206)( 95,204)( 96,203)( 97,201)( 98,202)
( 99,207)(100,208)(101,209)(102,196)(103,197)(104,195)(105,194)(106,192)
(107,193)(108,198)(109,199)(110,200);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(111,138)(112,140)(113,139)(114,144)(115,146)(116,145)(117,141)(118,143)
(119,142)(120,147)(121,149)(122,148)(123,153)(124,155)(125,154)(126,150)
(127,152)(128,151)(129,156)(130,158)(131,157)(132,162)(133,164)(134,163)
(135,159)(136,161)(137,160)(165,192)(166,194)(167,193)(168,198)(169,200)
(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)(177,207)
(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)(185,211)
(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);;
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*s1*s0*s1,
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 60, 63)( 61, 64)( 62, 65)( 66, 75)( 67, 76)( 68, 77)( 69, 81)
( 70, 82)( 71, 83)( 72, 78)( 73, 79)( 74, 80)( 87, 90)( 88, 91)( 89, 92)
( 93,102)( 94,103)( 95,104)( 96,108)( 97,109)( 98,110)( 99,105)(100,106)
(101,107)(111,192)(112,193)(113,194)(114,198)(115,199)(116,200)(117,195)
(118,196)(119,197)(120,210)(121,211)(122,212)(123,216)(124,217)(125,218)
(126,213)(127,214)(128,215)(129,201)(130,202)(131,203)(132,207)(133,208)
(134,209)(135,204)(136,205)(137,206)(138,165)(139,166)(140,167)(141,171)
(142,172)(143,173)(144,168)(145,169)(146,170)(147,183)(148,184)(149,185)
(150,189)(151,190)(152,191)(153,186)(154,187)(155,188)(156,174)(157,175)
(158,176)(159,180)(160,181)(161,182)(162,177)(163,178)(164,179);
s2 := Sym(218)!(  3,133)(  4,134)(  5,132)(  6,131)(  7,129)(  8,130)(  9,135)
( 10,136)( 11,137)( 12,124)( 13,125)( 14,123)( 15,122)( 16,120)( 17,121)
( 18,126)( 19,127)( 20,128)( 21,115)( 22,116)( 23,114)( 24,113)( 25,111)
( 26,112)( 27,117)( 28,118)( 29,119)( 30,160)( 31,161)( 32,159)( 33,158)
( 34,156)( 35,157)( 36,162)( 37,163)( 38,164)( 39,151)( 40,152)( 41,150)
( 42,149)( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,142)( 49,143)
( 50,141)( 51,140)( 52,138)( 53,139)( 54,144)( 55,145)( 56,146)( 57,187)
( 58,188)( 59,186)( 60,185)( 61,183)( 62,184)( 63,189)( 64,190)( 65,191)
( 66,178)( 67,179)( 68,177)( 69,176)( 70,174)( 71,175)( 72,180)( 73,181)
( 74,182)( 75,169)( 76,170)( 77,168)( 78,167)( 79,165)( 80,166)( 81,171)
( 82,172)( 83,173)( 84,214)( 85,215)( 86,213)( 87,212)( 88,210)( 89,211)
( 90,216)( 91,217)( 92,218)( 93,205)( 94,206)( 95,204)( 96,203)( 97,201)
( 98,202)( 99,207)(100,208)(101,209)(102,196)(103,197)(104,195)(105,194)
(106,192)(107,193)(108,198)(109,199)(110,200);
s3 := Sym(218)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(111,138)(112,140)(113,139)(114,144)(115,146)(116,145)(117,141)
(118,143)(119,142)(120,147)(121,149)(122,148)(123,153)(124,155)(125,154)
(126,150)(127,152)(128,151)(129,156)(130,158)(131,157)(132,162)(133,164)
(134,163)(135,159)(136,161)(137,160)(165,192)(166,194)(167,193)(168,198)
(169,200)(170,199)(171,195)(172,197)(173,196)(174,201)(175,203)(176,202)
(177,207)(178,209)(179,208)(180,204)(181,206)(182,205)(183,210)(184,212)
(185,211)(186,216)(187,218)(188,217)(189,213)(190,215)(191,214);
poly := sub<Sym(218)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s3*s1*s2*s3*s2*s1*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s1*s2*s3*s2*s1*s2,
s3*s1*s2*s3*s2*s1*s2*s3*s2*s3*s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

```

to this polytope