Polytope of Type {2,24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,6}*1728a
if this polytope has a name.
Group : SmallGroup(1728,15888)
Rank : 4
Schlafli Type : {2,24,6}
Number of vertices, edges, etc : 2, 72, 216, 18
Order of s₀s₁s₂s₃ : 24
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 : {2,12,6}*864a
   3-fold quotients : {2,24,6}*576b
   4-fold quotients : {2,6,6}*432c
   6-fold quotients : {2,12,6}*288b
   8-fold quotients : {2,3,6}*216
   9-fold quotients : {2,24,2}*192
   12-fold quotients : {2,6,6}*144c
   18-fold quotients : {2,12,2}*96
   24-fold quotients : {2,3,6}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {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 := (  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)( 16, 28)
( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)( 39, 48)
( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)( 47, 53)
( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)( 63, 87)( 64, 88)
( 65, 89)( 66,102)( 67,103)( 68,104)( 69,108)( 70,109)( 71,110)( 72,105)
( 73,106)( 74,107)( 75, 93)( 76, 94)( 77, 95)( 78, 99)( 79,100)( 80,101)
( 81, 96)( 82, 97)( 83, 98)(111,165)(112,166)(113,167)(114,171)(115,172)
(116,173)(117,168)(118,169)(119,170)(120,183)(121,184)(122,185)(123,189)
(124,190)(125,191)(126,186)(127,187)(128,188)(129,174)(130,175)(131,176)
(132,180)(133,181)(134,182)(135,177)(136,178)(137,179)(138,192)(139,193)
(140,194)(141,198)(142,199)(143,200)(144,195)(145,196)(146,197)(147,210)
(148,211)(149,212)(150,216)(151,217)(152,218)(153,213)(154,214)(155,215)
(156,201)(157,202)(158,203)(159,207)(160,208)(161,209)(162,204)(163,205)
(164,206);;
s2 := (  3,133)(  4,134)(  5,132)(  6,131)(  7,129)(  8,130)(  9,135)( 10,136)
( 11,137)( 12,124)( 13,125)( 14,123)( 15,122)( 16,120)( 17,121)( 18,126)
( 19,127)( 20,128)( 21,115)( 22,116)( 23,114)( 24,113)( 25,111)( 26,112)
( 27,117)( 28,118)( 29,119)( 30,160)( 31,161)( 32,159)( 33,158)( 34,156)
( 35,157)( 36,162)( 37,163)( 38,164)( 39,151)( 40,152)( 41,150)( 42,149)
( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,142)( 49,143)( 50,141)
( 51,140)( 52,138)( 53,139)( 54,144)( 55,145)( 56,146)( 57,214)( 58,215)
( 59,213)( 60,212)( 61,210)( 62,211)( 63,216)( 64,217)( 65,218)( 66,205)
( 67,206)( 68,204)( 69,203)( 70,201)( 71,202)( 72,207)( 73,208)( 74,209)
( 75,196)( 76,197)( 77,195)( 78,194)( 79,192)( 80,193)( 81,198)( 82,199)
( 83,200)( 84,187)( 85,188)( 86,186)( 87,185)( 88,183)( 89,184)( 90,189)
( 91,190)( 92,191)( 93,178)( 94,179)( 95,177)( 96,176)( 97,174)( 98,175)
( 99,180)(100,181)(101,182)(102,169)(103,170)(104,168)(105,167)(106,165)
(107,166)(108,171)(109,172)(110,173);;
s3 := (  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)( 17, 19)
( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)( 35, 37)
( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)( 53, 55)
( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)( 71, 73)
( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)( 89, 91)
( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)(107,109)
(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)(125,127)
(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)(143,145)
(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)(161,163)
(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)(179,181)
(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)(197,199)
(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)(215,217);;
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, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*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*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(218)!(1,2);
s1 := Sym(218)!(  6,  9)(  7, 10)(  8, 11)( 12, 21)( 13, 22)( 14, 23)( 15, 27)
( 16, 28)( 17, 29)( 18, 24)( 19, 25)( 20, 26)( 33, 36)( 34, 37)( 35, 38)
( 39, 48)( 40, 49)( 41, 50)( 42, 54)( 43, 55)( 44, 56)( 45, 51)( 46, 52)
( 47, 53)( 57, 84)( 58, 85)( 59, 86)( 60, 90)( 61, 91)( 62, 92)( 63, 87)
( 64, 88)( 65, 89)( 66,102)( 67,103)( 68,104)( 69,108)( 70,109)( 71,110)
( 72,105)( 73,106)( 74,107)( 75, 93)( 76, 94)( 77, 95)( 78, 99)( 79,100)
( 80,101)( 81, 96)( 82, 97)( 83, 98)(111,165)(112,166)(113,167)(114,171)
(115,172)(116,173)(117,168)(118,169)(119,170)(120,183)(121,184)(122,185)
(123,189)(124,190)(125,191)(126,186)(127,187)(128,188)(129,174)(130,175)
(131,176)(132,180)(133,181)(134,182)(135,177)(136,178)(137,179)(138,192)
(139,193)(140,194)(141,198)(142,199)(143,200)(144,195)(145,196)(146,197)
(147,210)(148,211)(149,212)(150,216)(151,217)(152,218)(153,213)(154,214)
(155,215)(156,201)(157,202)(158,203)(159,207)(160,208)(161,209)(162,204)
(163,205)(164,206);
s2 := Sym(218)!(  3,133)(  4,134)(  5,132)(  6,131)(  7,129)(  8,130)(  9,135)
( 10,136)( 11,137)( 12,124)( 13,125)( 14,123)( 15,122)( 16,120)( 17,121)
( 18,126)( 19,127)( 20,128)( 21,115)( 22,116)( 23,114)( 24,113)( 25,111)
( 26,112)( 27,117)( 28,118)( 29,119)( 30,160)( 31,161)( 32,159)( 33,158)
( 34,156)( 35,157)( 36,162)( 37,163)( 38,164)( 39,151)( 40,152)( 41,150)
( 42,149)( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,142)( 49,143)
( 50,141)( 51,140)( 52,138)( 53,139)( 54,144)( 55,145)( 56,146)( 57,214)
( 58,215)( 59,213)( 60,212)( 61,210)( 62,211)( 63,216)( 64,217)( 65,218)
( 66,205)( 67,206)( 68,204)( 69,203)( 70,201)( 71,202)( 72,207)( 73,208)
( 74,209)( 75,196)( 76,197)( 77,195)( 78,194)( 79,192)( 80,193)( 81,198)
( 82,199)( 83,200)( 84,187)( 85,188)( 86,186)( 87,185)( 88,183)( 89,184)
( 90,189)( 91,190)( 92,191)( 93,178)( 94,179)( 95,177)( 96,176)( 97,174)
( 98,175)( 99,180)(100,181)(101,182)(102,169)(103,170)(104,168)(105,167)
(106,165)(107,166)(108,171)(109,172)(110,173);
s3 := Sym(218)!(  4,  5)(  6,  9)(  7, 11)(  8, 10)( 13, 14)( 15, 18)( 16, 20)
( 17, 19)( 22, 23)( 24, 27)( 25, 29)( 26, 28)( 31, 32)( 33, 36)( 34, 38)
( 35, 37)( 40, 41)( 42, 45)( 43, 47)( 44, 46)( 49, 50)( 51, 54)( 52, 56)
( 53, 55)( 58, 59)( 60, 63)( 61, 65)( 62, 64)( 67, 68)( 69, 72)( 70, 74)
( 71, 73)( 76, 77)( 78, 81)( 79, 83)( 80, 82)( 85, 86)( 87, 90)( 88, 92)
( 89, 91)( 94, 95)( 96, 99)( 97,101)( 98,100)(103,104)(105,108)(106,110)
(107,109)(112,113)(114,117)(115,119)(116,118)(121,122)(123,126)(124,128)
(125,127)(130,131)(132,135)(133,137)(134,136)(139,140)(141,144)(142,146)
(143,145)(148,149)(150,153)(151,155)(152,154)(157,158)(159,162)(160,164)
(161,163)(166,167)(168,171)(169,173)(170,172)(175,176)(177,180)(178,182)
(179,181)(184,185)(186,189)(187,191)(188,190)(193,194)(195,198)(196,200)
(197,199)(202,203)(204,207)(205,209)(206,208)(211,212)(213,216)(214,218)
(215,217);
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, s1*s2*s3*s2*s1*s2*s1*s2*s3*s2*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s3*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*s2*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;

to this polytope