Polytope of Type {6,6,24}

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

References : None.
to this polytope