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# Polytope of Type {4,36,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,36,6}*1728a
Also Known As : {{4,36|2},{36,6|2}}. if this polytope has another name.
Group : SmallGroup(1728,14461)
Rank : 4
Schlafli Type : {4,36,6}
Number of vertices, edges, etc : 4, 72, 108, 6
Order of s0s1s2s3 : 36
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 : {2,36,6}*864a, {4,18,6}*864a
3-fold quotients : {4,36,2}*576a, {4,12,6}*576a
4-fold quotients : {2,18,6}*432a
6-fold quotients : {2,36,2}*288, {4,18,2}*288a, {2,12,6}*288a, {4,6,6}*288a
9-fold quotients : {4,12,2}*192a, {4,4,6}*192
12-fold quotients : {2,18,2}*144, {2,6,6}*144a
18-fold quotients : {2,12,2}*96, {2,4,6}*96a, {4,2,6}*96, {4,6,2}*96a
24-fold quotients : {2,9,2}*72
27-fold quotients : {4,4,2}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
54-fold quotients : {2,4,2}*32, {4,2,2}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (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 := (  1,109)(  2,111)(  3,110)(  4,117)(  5,116)(  6,115)(  7,114)(  8,113)
(  9,112)( 10,118)( 11,120)( 12,119)( 13,126)( 14,125)( 15,124)( 16,123)
( 17,122)( 18,121)( 19,127)( 20,129)( 21,128)( 22,135)( 23,134)( 24,133)
( 25,132)( 26,131)( 27,130)( 28,136)( 29,138)( 30,137)( 31,144)( 32,143)
( 33,142)( 34,141)( 35,140)( 36,139)( 37,145)( 38,147)( 39,146)( 40,153)
( 41,152)( 42,151)( 43,150)( 44,149)( 45,148)( 46,154)( 47,156)( 48,155)
( 49,162)( 50,161)( 51,160)( 52,159)( 53,158)( 54,157)( 55,163)( 56,165)
( 57,164)( 58,171)( 59,170)( 60,169)( 61,168)( 62,167)( 63,166)( 64,172)
( 65,174)( 66,173)( 67,180)( 68,179)( 69,178)( 70,177)( 71,176)( 72,175)
( 73,181)( 74,183)( 75,182)( 76,189)( 77,188)( 78,187)( 79,186)( 80,185)
( 81,184)( 82,190)( 83,192)( 84,191)( 85,198)( 86,197)( 87,196)( 88,195)
( 89,194)( 90,193)( 91,199)( 92,201)( 93,200)( 94,207)( 95,206)( 96,205)
( 97,204)( 98,203)( 99,202)(100,208)(101,210)(102,209)(103,216)(104,215)
(105,214)(106,213)(107,212)(108,211);;
s2 := (  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 22)( 11, 24)( 12, 23)( 13, 19)
( 14, 21)( 15, 20)( 16, 27)( 17, 26)( 18, 25)( 28, 31)( 29, 33)( 30, 32)
( 34, 36)( 37, 49)( 38, 51)( 39, 50)( 40, 46)( 41, 48)( 42, 47)( 43, 54)
( 44, 53)( 45, 52)( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 76)( 65, 78)
( 66, 77)( 67, 73)( 68, 75)( 69, 74)( 70, 81)( 71, 80)( 72, 79)( 82, 85)
( 83, 87)( 84, 86)( 88, 90)( 91,103)( 92,105)( 93,104)( 94,100)( 95,102)
( 96,101)( 97,108)( 98,107)( 99,106)(109,139)(110,141)(111,140)(112,136)
(113,138)(114,137)(115,144)(116,143)(117,142)(118,157)(119,159)(120,158)
(121,154)(122,156)(123,155)(124,162)(125,161)(126,160)(127,148)(128,150)
(129,149)(130,145)(131,147)(132,146)(133,153)(134,152)(135,151)(163,193)
(164,195)(165,194)(166,190)(167,192)(168,191)(169,198)(170,197)(171,196)
(172,211)(173,213)(174,212)(175,208)(176,210)(177,209)(178,216)(179,215)
(180,214)(181,202)(182,204)(183,203)(184,199)(185,201)(186,200)(187,207)
(188,206)(189,205);;
s3 := (  1, 10)(  2, 11)(  3, 12)(  4, 13)(  5, 14)(  6, 15)(  7, 16)(  8, 17)
(  9, 18)( 28, 37)( 29, 38)( 30, 39)( 31, 40)( 32, 41)( 33, 42)( 34, 43)
( 35, 44)( 36, 45)( 55, 64)( 56, 65)( 57, 66)( 58, 67)( 59, 68)( 60, 69)
( 61, 70)( 62, 71)( 63, 72)( 82, 91)( 83, 92)( 84, 93)( 85, 94)( 86, 95)
( 87, 96)( 88, 97)( 89, 98)( 90, 99)(109,118)(110,119)(111,120)(112,121)
(113,122)(114,123)(115,124)(116,125)(117,126)(136,145)(137,146)(138,147)
(139,148)(140,149)(141,150)(142,151)(143,152)(144,153)(163,172)(164,173)
(165,174)(166,175)(167,176)(168,177)(169,178)(170,179)(171,180)(190,199)
(191,200)(192,201)(193,202)(194,203)(195,204)(196,205)(197,206)(198,207);;
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, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope