Polytope of Type {24,6,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6,6}*1728b
Also Known As : {{24,6|2},{6,6|2}}. if this polytope has another name.
Group : SmallGroup(1728,33799)
Rank : 4
Schlafli Type : {24,6,6}
Number of vertices, edges, etc : 24, 72, 18, 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 : {12,6,6}*864b
3-fold quotients : {24,2,6}*576, {24,6,2}*576a, {8,6,6}*576a
4-fold quotients : {6,6,6}*432b
6-fold quotients : {24,2,3}*288, {12,2,6}*288, {12,6,2}*288a, {4,6,6}*288a
9-fold quotients : {24,2,2}*192, {8,2,6}*192, {8,6,2}*192
12-fold quotients : {12,2,3}*144, {2,6,6}*144a, {6,2,6}*144, {6,6,2}*144a
18-fold quotients : {8,2,3}*96, {12,2,2}*96, {4,2,6}*96, {4,6,2}*96a
24-fold quotients : {3,2,6}*72, {6,2,3}*72
27-fold quotients : {8,2,2}*64
36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48, {6,2,2}*48
48-fold quotients : {3,2,3}*36
54-fold quotients : {4,2,2}*32
72-fold quotients : {2,2,3}*24, {2,3,2}*24, {3,2,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Irregular Quotients (of which this is a minimal cover):
None.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64, 91)( 65, 93)( 66, 92)( 67, 94)( 68, 96)( 69, 95)( 70, 97)( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)( 76,103)( 77,105)( 78,104)( 79,106)( 80,108)( 81,107)(109,163)(110,165)(111,164)(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)(119,174)(120,173)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,181)(128,183)(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)(151,205)(152,207)(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)(160,214)(161,216)(162,215);;
s1 := ( 1,110)( 2,109)( 3,111)( 4,116)( 5,115)( 6,117)( 7,113)( 8,112)( 9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)( 40,152)( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)( 48,156)( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)( 56,190)( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,200)( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)( 72,204)( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)( 96,180)( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)(104,187)(105,189)(106,185)(107,184)(108,186);;
s2 := ( 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);;
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, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(216)!( 2, 3)( 5, 6)( 8, 9)( 11, 12)( 14, 15)( 17, 18)( 20, 21)( 23, 24)( 26, 27)( 29, 30)( 32, 33)( 35, 36)( 38, 39)( 41, 42)( 44, 45)( 47, 48)( 50, 51)( 53, 54)( 55, 82)( 56, 84)( 57, 83)( 58, 85)( 59, 87)( 60, 86)( 61, 88)( 62, 90)( 63, 89)( 64, 91)( 65, 93)( 66, 92)( 67, 94)( 68, 96)( 69, 95)( 70, 97)( 71, 99)( 72, 98)( 73,100)( 74,102)( 75,101)( 76,103)( 77,105)( 78,104)( 79,106)( 80,108)( 81,107)(109,163)(110,165)(111,164)(112,166)(113,168)(114,167)(115,169)(116,171)(117,170)(118,172)(119,174)(120,173)(121,175)(122,177)(123,176)(124,178)(125,180)(126,179)(127,181)(128,183)(129,182)(130,184)(131,186)(132,185)(133,187)(134,189)(135,188)(136,190)(137,192)(138,191)(139,193)(140,195)(141,194)(142,196)(143,198)(144,197)(145,199)(146,201)(147,200)(148,202)(149,204)(150,203)(151,205)(152,207)(153,206)(154,208)(155,210)(156,209)(157,211)(158,213)(159,212)(160,214)(161,216)(162,215);
s1 := Sym(216)!( 1,110)( 2,109)( 3,111)( 4,116)( 5,115)( 6,117)( 7,113)( 8,112)( 9,114)( 10,119)( 11,118)( 12,120)( 13,125)( 14,124)( 15,126)( 16,122)( 17,121)( 18,123)( 19,128)( 20,127)( 21,129)( 22,134)( 23,133)( 24,135)( 25,131)( 26,130)( 27,132)( 28,137)( 29,136)( 30,138)( 31,143)( 32,142)( 33,144)( 34,140)( 35,139)( 36,141)( 37,146)( 38,145)( 39,147)( 40,152)( 41,151)( 42,153)( 43,149)( 44,148)( 45,150)( 46,155)( 47,154)( 48,156)( 49,161)( 50,160)( 51,162)( 52,158)( 53,157)( 54,159)( 55,191)( 56,190)( 57,192)( 58,197)( 59,196)( 60,198)( 61,194)( 62,193)( 63,195)( 64,200)( 65,199)( 66,201)( 67,206)( 68,205)( 69,207)( 70,203)( 71,202)( 72,204)( 73,209)( 74,208)( 75,210)( 76,215)( 77,214)( 78,216)( 79,212)( 80,211)( 81,213)( 82,164)( 83,163)( 84,165)( 85,170)( 86,169)( 87,171)( 88,167)( 89,166)( 90,168)( 91,173)( 92,172)( 93,174)( 94,179)( 95,178)( 96,180)( 97,176)( 98,175)( 99,177)(100,182)(101,181)(102,183)(103,188)(104,187)(105,189)(106,185)(107,184)(108,186);
s2 := 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);
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,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope