Polytope of Type {3,6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,24}*864a
if this polytope has a name.
Group : SmallGroup(864,814)
Rank : 4
Schlafli Type : {3,6,24}
Number of vertices, edges, etc : 3, 9, 72, 24
Order of s₀s₁s₂s₃ : 24
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,24,2} of size 1728
Vertex Figure Of :
   {2,3,6,24} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,6,12}*432a
   3-fold quotients : {3,2,24}*288
   4-fold quotients : {3,6,6}*216a
   6-fold quotients : {3,2,12}*144
   8-fold quotients : {3,6,3}*108
   9-fold quotients : {3,2,8}*96
   12-fold quotients : {3,2,6}*72
   18-fold quotients : {3,2,4}*48
   24-fold quotients : {3,2,3}*36
   36-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,48}*1728a, {6,6,24}*1728a
Permutation Representation (GAP) :

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