Polytope of Type {3,2,3,24}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,3,24}*1728
if this polytope has a name.
Group : SmallGroup(1728,46303)
Rank : 5
Schlafli Type : {3,2,3,24}
Number of vertices, edges, etc : 3, 3, 6, 72, 48
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 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 : {3,2,3,12}*864
3-fold quotients : {3,2,3,8}*576
6-fold quotients : {3,2,3,4}*288
8-fold quotients : {3,2,3,6}*216
12-fold quotients : {3,2,3,4}*144
24-fold quotients : {3,2,3,2}*72
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,220)( 5,221)( 6,224)( 7,225)( 8,222)( 9,223)( 10,227)( 11,226)
( 12,236)( 13,237)( 14,240)( 15,241)( 16,238)( 17,239)( 18,243)( 19,242)
( 20,228)( 21,229)( 22,232)( 23,233)( 24,230)( 25,231)( 26,235)( 27,234)
( 28,268)( 29,269)( 30,272)( 31,273)( 32,270)( 33,271)( 34,275)( 35,274)
( 36,284)( 37,285)( 38,288)( 39,289)( 40,286)( 41,287)( 42,291)( 43,290)
( 44,276)( 45,277)( 46,280)( 47,281)( 48,278)( 49,279)( 50,283)( 51,282)
( 52,244)( 53,245)( 54,248)( 55,249)( 56,246)( 57,247)( 58,251)( 59,250)
( 60,260)( 61,261)( 62,264)( 63,265)( 64,262)( 65,263)( 66,267)( 67,266)
( 68,252)( 69,253)( 70,256)( 71,257)( 72,254)( 73,255)( 74,259)( 75,258)
( 76,149)( 77,148)( 78,153)( 79,152)( 80,151)( 81,150)( 82,154)( 83,155)
( 84,165)( 85,164)( 86,169)( 87,168)( 88,167)( 89,166)( 90,170)( 91,171)
( 92,157)( 93,156)( 94,161)( 95,160)( 96,159)( 97,158)( 98,162)( 99,163)
(100,197)(101,196)(102,201)(103,200)(104,199)(105,198)(106,202)(107,203)
(108,213)(109,212)(110,217)(111,216)(112,215)(113,214)(114,218)(115,219)
(116,205)(117,204)(118,209)(119,208)(120,207)(121,206)(122,210)(123,211)
(124,173)(125,172)(126,177)(127,176)(128,175)(129,174)(130,178)(131,179)
(132,189)(133,188)(134,193)(135,192)(136,191)(137,190)(138,194)(139,195)
(140,181)(141,180)(142,185)(143,184)(144,183)(145,182)(146,186)(147,187);;
s3 := ( 4,252)( 5,253)( 6,255)( 7,254)( 8,258)( 9,259)( 10,256)( 11,257)
( 12,244)( 13,245)( 14,247)( 15,246)( 16,250)( 17,251)( 18,248)( 19,249)
( 20,260)( 21,261)( 22,263)( 23,262)( 24,266)( 25,267)( 26,264)( 27,265)
( 28,228)( 29,229)( 30,231)( 31,230)( 32,234)( 33,235)( 34,232)( 35,233)
( 36,220)( 37,221)( 38,223)( 39,222)( 40,226)( 41,227)( 42,224)( 43,225)
( 44,236)( 45,237)( 46,239)( 47,238)( 48,242)( 49,243)( 50,240)( 51,241)
( 52,276)( 53,277)( 54,279)( 55,278)( 56,282)( 57,283)( 58,280)( 59,281)
( 60,268)( 61,269)( 62,271)( 63,270)( 64,274)( 65,275)( 66,272)( 67,273)
( 68,284)( 69,285)( 70,287)( 71,286)( 72,290)( 73,291)( 74,288)( 75,289)
( 76,181)( 77,180)( 78,182)( 79,183)( 80,187)( 81,186)( 82,185)( 83,184)
( 84,173)( 85,172)( 86,174)( 87,175)( 88,179)( 89,178)( 90,177)( 91,176)
( 92,189)( 93,188)( 94,190)( 95,191)( 96,195)( 97,194)( 98,193)( 99,192)
(100,157)(101,156)(102,158)(103,159)(104,163)(105,162)(106,161)(107,160)
(108,149)(109,148)(110,150)(111,151)(112,155)(113,154)(114,153)(115,152)
(116,165)(117,164)(118,166)(119,167)(120,171)(121,170)(122,169)(123,168)
(124,205)(125,204)(126,206)(127,207)(128,211)(129,210)(130,209)(131,208)
(132,197)(133,196)(134,198)(135,199)(136,203)(137,202)(138,201)(139,200)
(140,213)(141,212)(142,214)(143,215)(144,219)(145,218)(146,217)(147,216);;
s4 := ( 4,154)( 5,155)( 6,152)( 7,153)( 8,151)( 9,150)( 10,149)( 11,148)
( 12,170)( 13,171)( 14,168)( 15,169)( 16,167)( 17,166)( 18,165)( 19,164)
( 20,162)( 21,163)( 22,160)( 23,161)( 24,159)( 25,158)( 26,157)( 27,156)
( 28,178)( 29,179)( 30,176)( 31,177)( 32,175)( 33,174)( 34,173)( 35,172)
( 36,194)( 37,195)( 38,192)( 39,193)( 40,191)( 41,190)( 42,189)( 43,188)
( 44,186)( 45,187)( 46,184)( 47,185)( 48,183)( 49,182)( 50,181)( 51,180)
( 52,202)( 53,203)( 54,200)( 55,201)( 56,199)( 57,198)( 58,197)( 59,196)
( 60,218)( 61,219)( 62,216)( 63,217)( 64,215)( 65,214)( 66,213)( 67,212)
( 68,210)( 69,211)( 70,208)( 71,209)( 72,207)( 73,206)( 74,205)( 75,204)
( 76,227)( 77,226)( 78,225)( 79,224)( 80,222)( 81,223)( 82,220)( 83,221)
( 84,243)( 85,242)( 86,241)( 87,240)( 88,238)( 89,239)( 90,236)( 91,237)
( 92,235)( 93,234)( 94,233)( 95,232)( 96,230)( 97,231)( 98,228)( 99,229)
(100,251)(101,250)(102,249)(103,248)(104,246)(105,247)(106,244)(107,245)
(108,267)(109,266)(110,265)(111,264)(112,262)(113,263)(114,260)(115,261)
(116,259)(117,258)(118,257)(119,256)(120,254)(121,255)(122,252)(123,253)
(124,275)(125,274)(126,273)(127,272)(128,270)(129,271)(130,268)(131,269)
(132,291)(133,290)(134,289)(135,288)(136,286)(137,287)(138,284)(139,285)
(140,283)(141,282)(142,281)(143,280)(144,278)(145,279)(146,276)(147,277);;
poly := Group([s0,s1,s2,s3,s4]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;; s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4,
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3,
s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s4*s3*s4*s3*s4*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(291)!(2,3);
s1 := Sym(291)!(1,2);
s2 := Sym(291)!( 4,220)( 5,221)( 6,224)( 7,225)( 8,222)( 9,223)( 10,227)
( 11,226)( 12,236)( 13,237)( 14,240)( 15,241)( 16,238)( 17,239)( 18,243)
( 19,242)( 20,228)( 21,229)( 22,232)( 23,233)( 24,230)( 25,231)( 26,235)
( 27,234)( 28,268)( 29,269)( 30,272)( 31,273)( 32,270)( 33,271)( 34,275)
( 35,274)( 36,284)( 37,285)( 38,288)( 39,289)( 40,286)( 41,287)( 42,291)
( 43,290)( 44,276)( 45,277)( 46,280)( 47,281)( 48,278)( 49,279)( 50,283)
( 51,282)( 52,244)( 53,245)( 54,248)( 55,249)( 56,246)( 57,247)( 58,251)
( 59,250)( 60,260)( 61,261)( 62,264)( 63,265)( 64,262)( 65,263)( 66,267)
( 67,266)( 68,252)( 69,253)( 70,256)( 71,257)( 72,254)( 73,255)( 74,259)
( 75,258)( 76,149)( 77,148)( 78,153)( 79,152)( 80,151)( 81,150)( 82,154)
( 83,155)( 84,165)( 85,164)( 86,169)( 87,168)( 88,167)( 89,166)( 90,170)
( 91,171)( 92,157)( 93,156)( 94,161)( 95,160)( 96,159)( 97,158)( 98,162)
( 99,163)(100,197)(101,196)(102,201)(103,200)(104,199)(105,198)(106,202)
(107,203)(108,213)(109,212)(110,217)(111,216)(112,215)(113,214)(114,218)
(115,219)(116,205)(117,204)(118,209)(119,208)(120,207)(121,206)(122,210)
(123,211)(124,173)(125,172)(126,177)(127,176)(128,175)(129,174)(130,178)
(131,179)(132,189)(133,188)(134,193)(135,192)(136,191)(137,190)(138,194)
(139,195)(140,181)(141,180)(142,185)(143,184)(144,183)(145,182)(146,186)
(147,187);
s3 := Sym(291)!( 4,252)( 5,253)( 6,255)( 7,254)( 8,258)( 9,259)( 10,256)
( 11,257)( 12,244)( 13,245)( 14,247)( 15,246)( 16,250)( 17,251)( 18,248)
( 19,249)( 20,260)( 21,261)( 22,263)( 23,262)( 24,266)( 25,267)( 26,264)
( 27,265)( 28,228)( 29,229)( 30,231)( 31,230)( 32,234)( 33,235)( 34,232)
( 35,233)( 36,220)( 37,221)( 38,223)( 39,222)( 40,226)( 41,227)( 42,224)
( 43,225)( 44,236)( 45,237)( 46,239)( 47,238)( 48,242)( 49,243)( 50,240)
( 51,241)( 52,276)( 53,277)( 54,279)( 55,278)( 56,282)( 57,283)( 58,280)
( 59,281)( 60,268)( 61,269)( 62,271)( 63,270)( 64,274)( 65,275)( 66,272)
( 67,273)( 68,284)( 69,285)( 70,287)( 71,286)( 72,290)( 73,291)( 74,288)
( 75,289)( 76,181)( 77,180)( 78,182)( 79,183)( 80,187)( 81,186)( 82,185)
( 83,184)( 84,173)( 85,172)( 86,174)( 87,175)( 88,179)( 89,178)( 90,177)
( 91,176)( 92,189)( 93,188)( 94,190)( 95,191)( 96,195)( 97,194)( 98,193)
( 99,192)(100,157)(101,156)(102,158)(103,159)(104,163)(105,162)(106,161)
(107,160)(108,149)(109,148)(110,150)(111,151)(112,155)(113,154)(114,153)
(115,152)(116,165)(117,164)(118,166)(119,167)(120,171)(121,170)(122,169)
(123,168)(124,205)(125,204)(126,206)(127,207)(128,211)(129,210)(130,209)
(131,208)(132,197)(133,196)(134,198)(135,199)(136,203)(137,202)(138,201)
(139,200)(140,213)(141,212)(142,214)(143,215)(144,219)(145,218)(146,217)
(147,216);
s4 := Sym(291)!( 4,154)( 5,155)( 6,152)( 7,153)( 8,151)( 9,150)( 10,149)
( 11,148)( 12,170)( 13,171)( 14,168)( 15,169)( 16,167)( 17,166)( 18,165)
( 19,164)( 20,162)( 21,163)( 22,160)( 23,161)( 24,159)( 25,158)( 26,157)
( 27,156)( 28,178)( 29,179)( 30,176)( 31,177)( 32,175)( 33,174)( 34,173)
( 35,172)( 36,194)( 37,195)( 38,192)( 39,193)( 40,191)( 41,190)( 42,189)
( 43,188)( 44,186)( 45,187)( 46,184)( 47,185)( 48,183)( 49,182)( 50,181)
( 51,180)( 52,202)( 53,203)( 54,200)( 55,201)( 56,199)( 57,198)( 58,197)
( 59,196)( 60,218)( 61,219)( 62,216)( 63,217)( 64,215)( 65,214)( 66,213)
( 67,212)( 68,210)( 69,211)( 70,208)( 71,209)( 72,207)( 73,206)( 74,205)
( 75,204)( 76,227)( 77,226)( 78,225)( 79,224)( 80,222)( 81,223)( 82,220)
( 83,221)( 84,243)( 85,242)( 86,241)( 87,240)( 88,238)( 89,239)( 90,236)
( 91,237)( 92,235)( 93,234)( 94,233)( 95,232)( 96,230)( 97,231)( 98,228)
( 99,229)(100,251)(101,250)(102,249)(103,248)(104,246)(105,247)(106,244)
(107,245)(108,267)(109,266)(110,265)(111,264)(112,262)(113,263)(114,260)
(115,261)(116,259)(117,258)(118,257)(119,256)(120,254)(121,255)(122,252)
(123,253)(124,275)(125,274)(126,273)(127,272)(128,270)(129,271)(130,268)
(131,269)(132,291)(133,290)(134,289)(135,288)(136,286)(137,287)(138,284)
(139,285)(140,283)(141,282)(142,281)(143,280)(144,278)(145,279)(146,276)
(147,277);
poly := sub<Sym(291)|s0,s1,s2,s3,s4>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1,
s2*s3*s2*s3*s2*s3, s4*s2*s3*s4*s3*s4*s3*s4*s3*s4*s2*s3*s4*s3*s4*s3*s4*s3 >;
to this polytope