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# Polytope of Type {2,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,12,4}*1728c
if this polytope has a name.
Group : SmallGroup(1728,46611)
Rank : 4
Schlafli Type : {2,12,4}
Number of vertices, edges, etc : 2, 108, 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 : {2,12,4}*864b
3-fold quotients : {2,4,4}*576
6-fold quotients : {2,4,4}*288
9-fold quotients : {2,12,4}*192a
12-fold quotients : {2,4,4}*144
18-fold quotients : {2,12,2}*96, {2,6,4}*96a
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 := (  4,  5)(  6, 24)(  7, 26)(  8, 25)(  9, 18)( 10, 20)( 11, 19)( 12, 21)
( 13, 23)( 14, 22)( 16, 17)( 28, 29)( 31, 32)( 33, 51)( 34, 53)( 35, 52)
( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 43, 44)( 55, 56)
( 58, 59)( 60, 78)( 61, 80)( 62, 79)( 63, 72)( 64, 74)( 65, 73)( 66, 75)
( 67, 77)( 68, 76)( 70, 71)( 82, 83)( 85, 86)( 87,105)( 88,107)( 89,106)
( 90, 99)( 91,101)( 92,100)( 93,102)( 94,104)( 95,103)( 97, 98)(109,110)
(111,192)(112,194)(113,193)(114,213)(115,215)(116,214)(117,207)(118,209)
(119,208)(120,210)(121,212)(122,211)(123,204)(124,206)(125,205)(126,198)
(127,200)(128,199)(129,201)(130,203)(131,202)(132,195)(133,197)(134,196)
(135,216)(136,218)(137,217)(138,165)(139,167)(140,166)(141,186)(142,188)
(143,187)(144,180)(145,182)(146,181)(147,183)(148,185)(149,184)(150,177)
(151,179)(152,178)(153,171)(154,173)(155,172)(156,174)(157,176)(158,175)
(159,168)(160,170)(161,169)(162,189)(163,191)(164,190);;
s2 := (  3,112)(  4,111)(  5,113)(  6,115)(  7,114)(  8,116)(  9,118)( 10,117)
( 11,119)( 12,136)( 13,135)( 14,137)( 15,130)( 16,129)( 17,131)( 18,133)
( 19,132)( 20,134)( 21,124)( 22,123)( 23,125)( 24,127)( 25,126)( 26,128)
( 27,121)( 28,120)( 29,122)( 30,139)( 31,138)( 32,140)( 33,142)( 34,141)
( 35,143)( 36,145)( 37,144)( 38,146)( 39,163)( 40,162)( 41,164)( 42,157)
( 43,156)( 44,158)( 45,160)( 46,159)( 47,161)( 48,151)( 49,150)( 50,152)
( 51,154)( 52,153)( 53,155)( 54,148)( 55,147)( 56,149)( 57,166)( 58,165)
( 59,167)( 60,169)( 61,168)( 62,170)( 63,172)( 64,171)( 65,173)( 66,190)
( 67,189)( 68,191)( 69,184)( 70,183)( 71,185)( 72,187)( 73,186)( 74,188)
( 75,178)( 76,177)( 77,179)( 78,181)( 79,180)( 80,182)( 81,175)( 82,174)
( 83,176)( 84,193)( 85,192)( 86,194)( 87,196)( 88,195)( 89,197)( 90,199)
( 91,198)( 92,200)( 93,217)( 94,216)( 95,218)( 96,211)( 97,210)( 98,212)
( 99,214)(100,213)(101,215)(102,205)(103,204)(104,206)(105,208)(106,207)
(107,209)(108,202)(109,201)(110,203);;
s3 := (  3, 69)(  4, 70)(  5, 71)(  6, 75)(  7, 76)(  8, 77)(  9, 63)( 10, 64)
( 11, 65)( 12, 78)( 13, 79)( 14, 80)( 15, 57)( 16, 58)( 17, 59)( 18, 72)
( 19, 73)( 20, 74)( 21, 60)( 22, 61)( 23, 62)( 24, 66)( 25, 67)( 26, 68)
( 27, 81)( 28, 82)( 29, 83)( 30, 96)( 31, 97)( 32, 98)( 33,102)( 34,103)
( 35,104)( 36, 90)( 37, 91)( 38, 92)( 39,105)( 40,106)( 41,107)( 42, 84)
( 43, 85)( 44, 86)( 45, 99)( 46,100)( 47,101)( 48, 87)( 49, 88)( 50, 89)
( 51, 93)( 52, 94)( 53, 95)( 54,108)( 55,109)( 56,110)(111,123)(112,124)
(113,125)(114,129)(115,130)(116,131)(120,132)(121,133)(122,134)(138,150)
(139,151)(140,152)(141,156)(142,157)(143,158)(147,159)(148,160)(149,161)
(165,177)(166,178)(167,179)(168,183)(169,184)(170,185)(174,186)(175,187)
(176,188)(192,204)(193,205)(194,206)(195,210)(196,211)(197,212)(201,213)
(202,214)(203,215);;
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,
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*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*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

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

```

to this polytope