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# Polytope of Type {12,6,6}

Atlas Canonical Name : {12,6,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,47874)
Rank : 4
Schlafli Type : {12,6,6}
Number of vertices, edges, etc : 24, 72, 36, 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) :
3-fold quotients : {4,6,6}*576a, {12,6,2}*576a
4-fold quotients : {6,6,6}*432b
6-fold quotients : {4,6,6}*288d, {12,6,2}*288d
9-fold quotients : {4,6,2}*192
12-fold quotients : {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
18-fold quotients : {4,3,2}*96, {4,6,2}*96b, {4,6,2}*96c
24-fold quotients : {3,2,6}*72, {6,2,3}*72
36-fold quotients : {4,3,2}*48, {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
48-fold quotients : {3,2,3}*36
72-fold quotients : {2,2,3}*24, {2,3,2}*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 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 27)( 14, 28)
( 15, 25)( 16, 26)( 17, 31)( 18, 32)( 19, 29)( 20, 30)( 21, 35)( 22, 36)
( 23, 33)( 24, 34)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 63)( 50, 64)( 51, 61)( 52, 62)( 53, 67)( 54, 68)( 55, 65)( 56, 66)
( 57, 71)( 58, 72)( 59, 69)( 60, 70)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)
( 91,101)( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,135)(122,136)(123,133)(124,134)
(125,139)(126,140)(127,137)(128,138)(129,143)(130,144)(131,141)(132,142)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,171)(158,172)
(159,169)(160,170)(161,175)(162,176)(163,173)(164,174)(165,179)(166,180)
(167,177)(168,178)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,207)(194,208)(195,205)(196,206)(197,211)(198,212)(199,209)(200,210)
(201,215)(202,216)(203,213)(204,214);;
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, 49)( 14, 52)( 15, 51)( 16, 50)
( 17, 57)( 18, 60)( 19, 59)( 20, 58)( 21, 53)( 22, 56)( 23, 55)( 24, 54)
( 25, 61)( 26, 64)( 27, 63)( 28, 62)( 29, 69)( 30, 72)( 31, 71)( 32, 70)
( 33, 65)( 34, 68)( 35, 67)( 36, 66)( 74, 76)( 77, 81)( 78, 84)( 79, 83)
( 80, 82)( 86, 88)( 89, 93)( 90, 96)( 91, 95)( 92, 94)( 98,100)(101,105)
(102,108)(103,107)(104,106)(109,145)(110,148)(111,147)(112,146)(113,153)
(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)(120,150)(121,157)
(122,160)(123,159)(124,158)(125,165)(126,168)(127,167)(128,166)(129,161)
(130,164)(131,163)(132,162)(133,169)(134,172)(135,171)(136,170)(137,177)
(138,180)(139,179)(140,178)(141,173)(142,176)(143,175)(144,174)(182,184)
(185,189)(186,192)(187,191)(188,190)(194,196)(197,201)(198,204)(199,203)
(200,202)(206,208)(209,213)(210,216)(211,215)(212,214);;
s3 := (  1,113)(  2,114)(  3,115)(  4,116)(  5,109)(  6,110)(  7,111)(  8,112)
(  9,117)( 10,118)( 11,119)( 12,120)( 13,125)( 14,126)( 15,127)( 16,128)
( 17,121)( 18,122)( 19,123)( 20,124)( 21,129)( 22,130)( 23,131)( 24,132)
( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)
( 33,141)( 34,142)( 35,143)( 36,144)( 37,149)( 38,150)( 39,151)( 40,152)
( 41,145)( 42,146)( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,156)
( 49,161)( 50,162)( 51,163)( 52,164)( 53,157)( 54,158)( 55,159)( 56,160)
( 57,165)( 58,166)( 59,167)( 60,168)( 61,173)( 62,174)( 63,175)( 64,176)
( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)
( 73,185)( 74,186)( 75,187)( 76,188)( 77,181)( 78,182)( 79,183)( 80,184)
( 81,189)( 82,190)( 83,191)( 84,192)( 85,197)( 86,198)( 87,199)( 88,200)
( 89,193)( 90,194)( 91,195)( 92,196)( 93,201)( 94,202)( 95,203)( 96,204)
( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)
(105,213)(106,214)(107,215)(108,216);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope