Polytope of Type {2,24,6}

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

Permutation Representation (Magma) :

s0 := Sym(218)!(1,2);
s1 := Sym(218)!(  4,  5)(  7,  8)( 10, 11)( 12, 21)( 13, 23)( 14, 22)( 15, 24)
( 16, 26)( 17, 25)( 18, 27)( 19, 29)( 20, 28)( 31, 32)( 34, 35)( 37, 38)
( 39, 48)( 40, 50)( 41, 49)( 42, 51)( 43, 53)( 44, 52)( 45, 54)( 46, 56)
( 47, 55)( 57, 84)( 58, 86)( 59, 85)( 60, 87)( 61, 89)( 62, 88)( 63, 90)
( 64, 92)( 65, 91)( 66,102)( 67,104)( 68,103)( 69,105)( 70,107)( 71,106)
( 72,108)( 73,110)( 74,109)( 75, 93)( 76, 95)( 77, 94)( 78, 96)( 79, 98)
( 80, 97)( 81, 99)( 82,101)( 83,100)(111,165)(112,167)(113,166)(114,168)
(115,170)(116,169)(117,171)(118,173)(119,172)(120,183)(121,185)(122,184)
(123,186)(124,188)(125,187)(126,189)(127,191)(128,190)(129,174)(130,176)
(131,175)(132,177)(133,179)(134,178)(135,180)(136,182)(137,181)(138,192)
(139,194)(140,193)(141,195)(142,197)(143,196)(144,198)(145,200)(146,199)
(147,210)(148,212)(149,211)(150,213)(151,215)(152,214)(153,216)(154,218)
(155,217)(156,201)(157,203)(158,202)(159,204)(160,206)(161,205)(162,207)
(163,209)(164,208);
s2 := Sym(218)!(  3,120)(  4,121)(  5,122)(  6,128)(  7,126)(  8,127)(  9,124)
( 10,125)( 11,123)( 12,111)( 13,112)( 14,113)( 15,119)( 16,117)( 17,118)
( 18,115)( 19,116)( 20,114)( 21,129)( 22,130)( 23,131)( 24,137)( 25,135)
( 26,136)( 27,133)( 28,134)( 29,132)( 30,147)( 31,148)( 32,149)( 33,155)
( 34,153)( 35,154)( 36,151)( 37,152)( 38,150)( 39,138)( 40,139)( 41,140)
( 42,146)( 43,144)( 44,145)( 45,142)( 46,143)( 47,141)( 48,156)( 49,157)
( 50,158)( 51,164)( 52,162)( 53,163)( 54,160)( 55,161)( 56,159)( 57,201)
( 58,202)( 59,203)( 60,209)( 61,207)( 62,208)( 63,205)( 64,206)( 65,204)
( 66,192)( 67,193)( 68,194)( 69,200)( 70,198)( 71,199)( 72,196)( 73,197)
( 74,195)( 75,210)( 76,211)( 77,212)( 78,218)( 79,216)( 80,217)( 81,214)
( 82,215)( 83,213)( 84,174)( 85,175)( 86,176)( 87,182)( 88,180)( 89,181)
( 90,178)( 91,179)( 92,177)( 93,165)( 94,166)( 95,167)( 96,173)( 97,171)
( 98,172)( 99,169)(100,170)(101,168)(102,183)(103,184)(104,185)(105,191)
(106,189)(107,190)(108,187)(109,188)(110,186);
s3 := Sym(218)!(  3,  6)(  4,  8)(  5,  7)( 10, 11)( 12, 15)( 13, 17)( 14, 16)
( 19, 20)( 21, 24)( 22, 26)( 23, 25)( 28, 29)( 30, 33)( 31, 35)( 32, 34)
( 37, 38)( 39, 42)( 40, 44)( 41, 43)( 46, 47)( 48, 51)( 49, 53)( 50, 52)
( 55, 56)( 57, 60)( 58, 62)( 59, 61)( 64, 65)( 66, 69)( 67, 71)( 68, 70)
( 73, 74)( 75, 78)( 76, 80)( 77, 79)( 82, 83)( 84, 87)( 85, 89)( 86, 88)
( 91, 92)( 93, 96)( 94, 98)( 95, 97)(100,101)(102,105)(103,107)(104,106)
(109,110)(111,114)(112,116)(113,115)(118,119)(120,123)(121,125)(122,124)
(127,128)(129,132)(130,134)(131,133)(136,137)(138,141)(139,143)(140,142)
(145,146)(147,150)(148,152)(149,151)(154,155)(156,159)(157,161)(158,160)
(163,164)(165,168)(166,170)(167,169)(172,173)(174,177)(175,179)(176,178)
(181,182)(183,186)(184,188)(185,187)(190,191)(192,195)(193,197)(194,196)
(199,200)(201,204)(202,206)(203,205)(208,209)(210,213)(211,215)(212,214)
(217,218);
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, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s3*s2*s1*s2*s3*s2*s3*s2*s1*s2*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*s1*s2 >;

to this polytope