Polytope of Type {6,24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24,6}*1728c
if this polytope has a name.
Group : SmallGroup(1728,33799)
Rank : 4
Schlafli Type : {6,24,6}
Number of vertices, edges, etc : 6, 72, 72, 6
Order of s₀s₁s₂s₃ : 24
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 : {6,12,6}*864c
   3-fold quotients : {2,24,6}*576a, {6,24,2}*576b
   4-fold quotients : {6,6,6}*432d
   6-fold quotients : {2,12,6}*288a, {6,12,2}*288b
   9-fold quotients : {2,24,2}*192, {2,8,6}*192
   12-fold quotients : {2,6,6}*144a, {6,6,2}*144b
   18-fold quotients : {2,12,2}*96, {2,4,6}*96a
   24-fold quotients : {6,3,2}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,4,2}*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 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(166,169)(167,170)
(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)(186,189)(193,196)
(194,197)(195,198)(202,205)(203,206)(204,207)(211,214)(212,215)(213,216);;
s1 := (  1,  4)(  2,  6)(  3,  5)(  8,  9)( 10, 13)( 11, 15)( 12, 14)( 17, 18)
( 19, 22)( 20, 24)( 21, 23)( 26, 27)( 28, 31)( 29, 33)( 30, 32)( 35, 36)
( 37, 40)( 38, 42)( 39, 41)( 44, 45)( 46, 49)( 47, 51)( 48, 50)( 53, 54)
( 55, 85)( 56, 87)( 57, 86)( 58, 82)( 59, 84)( 60, 83)( 61, 88)( 62, 90)
( 63, 89)( 64, 94)( 65, 96)( 66, 95)( 67, 91)( 68, 93)( 69, 92)( 70, 97)
( 71, 99)( 72, 98)( 73,103)( 74,105)( 75,104)( 76,100)( 77,102)( 78,101)
( 79,106)( 80,108)( 81,107)(109,166)(110,168)(111,167)(112,163)(113,165)
(114,164)(115,169)(116,171)(117,170)(118,175)(119,177)(120,176)(121,172)
(122,174)(123,173)(124,178)(125,180)(126,179)(127,184)(128,186)(129,185)
(130,181)(131,183)(132,182)(133,187)(134,189)(135,188)(136,193)(137,195)
(138,194)(139,190)(140,192)(141,191)(142,196)(143,198)(144,197)(145,202)
(146,204)(147,203)(148,199)(149,201)(150,200)(151,205)(152,207)(153,206)
(154,211)(155,213)(156,212)(157,208)(158,210)(159,209)(160,214)(161,216)
(162,215);;
s2 := (  1,110)(  2,109)(  3,111)(  4,116)(  5,115)(  6,117)(  7,113)(  8,112)
(  9,114)( 10,128)( 11,127)( 12,129)( 13,134)( 14,133)( 15,135)( 16,131)
( 17,130)( 18,132)( 19,119)( 20,118)( 21,120)( 22,125)( 23,124)( 24,126)
( 25,122)( 26,121)( 27,123)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)
( 33,144)( 34,140)( 35,139)( 36,141)( 37,155)( 38,154)( 39,156)( 40,161)
( 41,160)( 42,162)( 43,158)( 44,157)( 45,159)( 46,146)( 47,145)( 48,147)
( 49,152)( 50,151)( 51,153)( 52,149)( 53,148)( 54,150)( 55,191)( 56,190)
( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,209)
( 65,208)( 66,210)( 67,215)( 68,214)( 69,216)( 70,212)( 71,211)( 72,213)
( 73,200)( 74,199)( 75,201)( 76,206)( 77,205)( 78,207)( 79,203)( 80,202)
( 81,204)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)
( 89,166)( 90,168)( 91,182)( 92,181)( 93,183)( 94,188)( 95,187)( 96,189)
( 97,185)( 98,184)( 99,186)(100,173)(101,172)(102,174)(103,179)(104,178)
(105,180)(106,176)(107,175)(108,177);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
to this polytope