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# Polytope of Type {6,12,4}

Atlas Canonical Name : {6,12,4}*1728i
if this polytope has a name.
Group : SmallGroup(1728,30413)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 18, 108, 72, 4
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 : {6,12,4}*864a, {6,12,2}*864e
3-fold quotients : {6,4,4}*576
4-fold quotients : {6,12,2}*432a
6-fold quotients : {6,4,4}*288, {6,4,2}*288
12-fold quotients : {6,4,2}*144
27-fold quotients : {2,4,4}*64
54-fold quotients : {2,2,4}*32, {2,4,2}*32
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,109)(  2,110)(  3,111)(  4,117)(  5,115)(  6,116)(  7,113)(  8,114)
(  9,112)( 10,129)( 11,127)( 12,128)( 13,134)( 14,135)( 15,133)( 16,130)
( 17,131)( 18,132)( 19,119)( 20,120)( 21,118)( 22,124)( 23,125)( 24,126)
( 25,123)( 26,121)( 27,122)( 28,136)( 29,137)( 30,138)( 31,144)( 32,142)
( 33,143)( 34,140)( 35,141)( 36,139)( 37,156)( 38,154)( 39,155)( 40,161)
( 41,162)( 42,160)( 43,157)( 44,158)( 45,159)( 46,146)( 47,147)( 48,145)
( 49,151)( 50,152)( 51,153)( 52,150)( 53,148)( 54,149)( 55,163)( 56,164)
( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)
( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)
( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)
( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)
( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)
( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)
(105,207)(106,204)(107,202)(108,203);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4,  5)(  7, 26)(  8, 25)(  9, 27)( 13, 23)
( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)
( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)
( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)
( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)
( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102)
(109,118)(110,120)(111,119)(112,113)(115,134)(116,133)(117,135)(121,131)
(122,130)(123,132)(124,125)(128,129)(136,145)(137,147)(138,146)(139,140)
(142,161)(143,160)(144,162)(148,158)(149,157)(150,159)(151,152)(155,156)
(163,172)(164,174)(165,173)(166,167)(169,188)(170,187)(171,189)(175,185)
(176,184)(177,186)(178,179)(182,183)(190,199)(191,201)(192,200)(193,194)
(196,215)(197,214)(198,216)(202,212)(203,211)(204,213)(205,206)(209,210);;
s2 := (  1,  2)(  4,  6)(  8,  9)( 10, 26)( 11, 25)( 12, 27)( 13, 21)( 14, 20)
( 15, 19)( 16, 22)( 17, 24)( 18, 23)( 28, 29)( 31, 33)( 35, 36)( 37, 53)
( 38, 52)( 39, 54)( 40, 48)( 41, 47)( 42, 46)( 43, 49)( 44, 51)( 45, 50)
( 55, 83)( 56, 82)( 57, 84)( 58, 87)( 59, 86)( 60, 85)( 61, 88)( 62, 90)
( 63, 89)( 64,107)( 65,106)( 66,108)( 67,102)( 68,101)( 69,100)( 70,103)
( 71,105)( 72,104)( 73, 96)( 74, 95)( 75, 94)( 76, 97)( 77, 99)( 78, 98)
( 79, 92)( 80, 91)( 81, 93)(109,110)(112,114)(116,117)(118,134)(119,133)
(120,135)(121,129)(122,128)(123,127)(124,130)(125,132)(126,131)(136,137)
(139,141)(143,144)(145,161)(146,160)(147,162)(148,156)(149,155)(150,154)
(151,157)(152,159)(153,158)(163,191)(164,190)(165,192)(166,195)(167,194)
(168,193)(169,196)(170,198)(171,197)(172,215)(173,214)(174,216)(175,210)
(176,209)(177,208)(178,211)(179,213)(180,212)(181,204)(182,203)(183,202)
(184,205)(185,207)(186,206)(187,200)(188,199)(189,201);;
s3 := (  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);;
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,
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope