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# Polytope of Type {4,12,6}

Atlas Canonical Name : {4,12,6}*1728h
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {4,12,6}
Number of vertices, edges, etc : 4, 72, 108, 18
Order of s0s1s2s3 : 12
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 : {4,12,6}*864b, {2,12,6}*864e
3-fold quotients : {4,4,6}*576
4-fold quotients : {2,12,6}*432b
6-fold quotients : {4,4,6}*288, {2,4,6}*288
12-fold quotients : {2,4,6}*144
27-fold quotients : {4,4,2}*64
54-fold quotients : {2,4,2}*32, {4,2,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
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,  5)(  7,  9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)
( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)
( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)
( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)
( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)
( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)
( 79, 91)( 80, 93)( 81, 92)(110,111)(112,113)(115,117)(118,133)(119,135)
(120,134)(121,128)(122,127)(123,129)(124,132)(125,131)(126,130)(137,138)
(139,140)(142,144)(145,160)(146,162)(147,161)(148,155)(149,154)(150,156)
(151,159)(152,158)(153,157)(163,190)(164,192)(165,191)(166,194)(167,193)
(168,195)(169,198)(170,197)(171,196)(172,214)(173,216)(174,215)(175,209)
(176,208)(177,210)(178,213)(179,212)(180,211)(181,203)(182,202)(183,204)
(184,207)(185,206)(186,205)(187,199)(188,201)(189,200);;
s2 := (  1,  2)(  4, 17)(  5, 16)(  6, 18)(  7, 22)(  8, 24)(  9, 23)( 10, 12)
( 13, 27)( 14, 26)( 15, 25)( 20, 21)( 28, 29)( 31, 44)( 32, 43)( 33, 45)
( 34, 49)( 35, 51)( 36, 50)( 37, 39)( 40, 54)( 41, 53)( 42, 52)( 47, 48)
( 55, 56)( 58, 71)( 59, 70)( 60, 72)( 61, 76)( 62, 78)( 63, 77)( 64, 66)
( 67, 81)( 68, 80)( 69, 79)( 74, 75)( 82, 83)( 85, 98)( 86, 97)( 87, 99)
( 88,103)( 89,105)( 90,104)( 91, 93)( 94,108)( 95,107)( 96,106)(101,102)
(109,110)(112,125)(113,124)(114,126)(115,130)(116,132)(117,131)(118,120)
(121,135)(122,134)(123,133)(128,129)(136,137)(139,152)(140,151)(141,153)
(142,157)(143,159)(144,158)(145,147)(148,162)(149,161)(150,160)(155,156)
(163,164)(166,179)(167,178)(168,180)(169,184)(170,186)(171,185)(172,174)
(175,189)(176,188)(177,187)(182,183)(190,191)(193,206)(194,205)(195,207)
(196,211)(197,213)(198,212)(199,201)(202,216)(203,215)(204,214)(209,210);;
s3 := (  1,116)(  2,117)(  3,115)(  4,112)(  5,113)(  6,114)(  7,111)(  8,109)
(  9,110)( 10,133)( 11,134)( 12,135)( 13,132)( 14,130)( 15,131)( 16,128)
( 17,129)( 18,127)( 19,126)( 20,124)( 21,125)( 22,122)( 23,123)( 24,121)
( 25,118)( 26,119)( 27,120)( 28,143)( 29,144)( 30,142)( 31,139)( 32,140)
( 33,141)( 34,138)( 35,136)( 36,137)( 37,160)( 38,161)( 39,162)( 40,159)
( 41,157)( 42,158)( 43,155)( 44,156)( 45,154)( 46,153)( 47,151)( 48,152)
( 49,149)( 50,150)( 51,148)( 52,145)( 53,146)( 54,147)( 55,170)( 56,171)
( 57,169)( 58,166)( 59,167)( 60,168)( 61,165)( 62,163)( 63,164)( 64,187)
( 65,188)( 66,189)( 67,186)( 68,184)( 69,185)( 70,182)( 71,183)( 72,181)
( 73,180)( 74,178)( 75,179)( 76,176)( 77,177)( 78,175)( 79,172)( 80,173)
( 81,174)( 82,197)( 83,198)( 84,196)( 85,193)( 86,194)( 87,195)( 88,192)
( 89,190)( 90,191)( 91,214)( 92,215)( 93,216)( 94,213)( 95,211)( 96,212)
( 97,209)( 98,210)( 99,208)(100,207)(101,205)(102,206)(103,203)(104,204)
(105,202)(106,199)(107,200)(108,201);;
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s1*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,  5)(  7,  9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)
( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)
( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)
( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)
( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)
( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)
( 78, 97)( 79, 91)( 80, 93)( 81, 92)(110,111)(112,113)(115,117)(118,133)
(119,135)(120,134)(121,128)(122,127)(123,129)(124,132)(125,131)(126,130)
(137,138)(139,140)(142,144)(145,160)(146,162)(147,161)(148,155)(149,154)
(150,156)(151,159)(152,158)(153,157)(163,190)(164,192)(165,191)(166,194)
(167,193)(168,195)(169,198)(170,197)(171,196)(172,214)(173,216)(174,215)
(175,209)(176,208)(177,210)(178,213)(179,212)(180,211)(181,203)(182,202)
(183,204)(184,207)(185,206)(186,205)(187,199)(188,201)(189,200);
s2 := Sym(216)!(  1,  2)(  4, 17)(  5, 16)(  6, 18)(  7, 22)(  8, 24)(  9, 23)
( 10, 12)( 13, 27)( 14, 26)( 15, 25)( 20, 21)( 28, 29)( 31, 44)( 32, 43)
( 33, 45)( 34, 49)( 35, 51)( 36, 50)( 37, 39)( 40, 54)( 41, 53)( 42, 52)
( 47, 48)( 55, 56)( 58, 71)( 59, 70)( 60, 72)( 61, 76)( 62, 78)( 63, 77)
( 64, 66)( 67, 81)( 68, 80)( 69, 79)( 74, 75)( 82, 83)( 85, 98)( 86, 97)
( 87, 99)( 88,103)( 89,105)( 90,104)( 91, 93)( 94,108)( 95,107)( 96,106)
(101,102)(109,110)(112,125)(113,124)(114,126)(115,130)(116,132)(117,131)
(118,120)(121,135)(122,134)(123,133)(128,129)(136,137)(139,152)(140,151)
(141,153)(142,157)(143,159)(144,158)(145,147)(148,162)(149,161)(150,160)
(155,156)(163,164)(166,179)(167,178)(168,180)(169,184)(170,186)(171,185)
(172,174)(175,189)(176,188)(177,187)(182,183)(190,191)(193,206)(194,205)
(195,207)(196,211)(197,213)(198,212)(199,201)(202,216)(203,215)(204,214)
(209,210);
s3 := Sym(216)!(  1,116)(  2,117)(  3,115)(  4,112)(  5,113)(  6,114)(  7,111)
(  8,109)(  9,110)( 10,133)( 11,134)( 12,135)( 13,132)( 14,130)( 15,131)
( 16,128)( 17,129)( 18,127)( 19,126)( 20,124)( 21,125)( 22,122)( 23,123)
( 24,121)( 25,118)( 26,119)( 27,120)( 28,143)( 29,144)( 30,142)( 31,139)
( 32,140)( 33,141)( 34,138)( 35,136)( 36,137)( 37,160)( 38,161)( 39,162)
( 40,159)( 41,157)( 42,158)( 43,155)( 44,156)( 45,154)( 46,153)( 47,151)
( 48,152)( 49,149)( 50,150)( 51,148)( 52,145)( 53,146)( 54,147)( 55,170)
( 56,171)( 57,169)( 58,166)( 59,167)( 60,168)( 61,165)( 62,163)( 63,164)
( 64,187)( 65,188)( 66,189)( 67,186)( 68,184)( 69,185)( 70,182)( 71,183)
( 72,181)( 73,180)( 74,178)( 75,179)( 76,176)( 77,177)( 78,175)( 79,172)
( 80,173)( 81,174)( 82,197)( 83,198)( 84,196)( 85,193)( 86,194)( 87,195)
( 88,192)( 89,190)( 90,191)( 91,214)( 92,215)( 93,216)( 94,213)( 95,211)
( 96,212)( 97,209)( 98,210)( 99,208)(100,207)(101,205)(102,206)(103,203)
(104,204)(105,202)(106,199)(107,200)(108,201);
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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s3*s2*s1*s2*s1*s2*s3*s2*s3*s2*s1*s2 >;

```
References : None.
to this polytope