Polytope of Type {4,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,12}*1152h
if this polytope has a name.
Group : SmallGroup(1152,157864)
Rank : 4
Schlafli Type : {4,6,12}
Number of vertices, edges, etc : 4, 24, 72, 24
Order of s0s1s2s3 : 6
Order of s0s1s2s3s2s1 : 2
Special Properties :
Non-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 : {4,3,12}*576
3-fold quotients : {4,6,4}*384f
4-fold quotients : {4,6,6}*288f
6-fold quotients : {4,3,4}*192a, {4,6,4}*192f, {4,6,4}*192g
8-fold quotients : {4,3,6}*144
12-fold quotients : {4,6,2}*96b, {4,3,4}*96
24-fold quotients : {4,3,2}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,153)( 2,154)( 3,155)( 4,156)( 5,157)( 6,158)( 7,159)( 8,160)
( 9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)( 16,152)
( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)( 24,176)
( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)( 32,168)
( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)( 40,192)
( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)( 48,184)
( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)( 56,208)
( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)( 64,200)
( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)( 72,224)
( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)( 80,216)
( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)( 88,240)
( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)( 96,232)
( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)(104,256)
(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)(112,248)
(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)(120,272)
(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)(128,264)
(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)(136,288)
(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)(144,280);;
s1 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)( 50, 98)
( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)( 58,110)
( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)( 66,130)
( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)( 74,142)
( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)( 82,114)
( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)( 90,126)
( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(147,148)(151,152)
(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)(164,179)
(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)(172,191)
(173,185)(174,186)(175,188)(176,187)(193,241)(194,242)(195,244)(196,243)
(197,245)(198,246)(199,248)(200,247)(201,253)(202,254)(203,256)(204,255)
(205,249)(206,250)(207,252)(208,251)(209,273)(210,274)(211,276)(212,275)
(213,277)(214,278)(215,280)(216,279)(217,285)(218,286)(219,288)(220,287)
(221,281)(222,282)(223,284)(224,283)(225,257)(226,258)(227,260)(228,259)
(229,261)(230,262)(231,264)(232,263)(233,269)(234,270)(235,272)(236,271)
(237,265)(238,266)(239,268)(240,267);;
s2 := ( 1,209)( 2,212)( 3,211)( 4,210)( 5,221)( 6,224)( 7,223)( 8,222)
( 9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)( 16,214)
( 17,193)( 18,196)( 19,195)( 20,194)( 21,205)( 22,208)( 23,207)( 24,206)
( 25,201)( 26,204)( 27,203)( 28,202)( 29,197)( 30,200)( 31,199)( 32,198)
( 33,225)( 34,228)( 35,227)( 36,226)( 37,237)( 38,240)( 39,239)( 40,238)
( 41,233)( 42,236)( 43,235)( 44,234)( 45,229)( 46,232)( 47,231)( 48,230)
( 49,161)( 50,164)( 51,163)( 52,162)( 53,173)( 54,176)( 55,175)( 56,174)
( 57,169)( 58,172)( 59,171)( 60,170)( 61,165)( 62,168)( 63,167)( 64,166)
( 65,145)( 66,148)( 67,147)( 68,146)( 69,157)( 70,160)( 71,159)( 72,158)
( 73,153)( 74,156)( 75,155)( 76,154)( 77,149)( 78,152)( 79,151)( 80,150)
( 81,177)( 82,180)( 83,179)( 84,178)( 85,189)( 86,192)( 87,191)( 88,190)
( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)( 96,182)
( 97,257)( 98,260)( 99,259)(100,258)(101,269)(102,272)(103,271)(104,270)
(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)(112,262)
(113,241)(114,244)(115,243)(116,242)(117,253)(118,256)(119,255)(120,254)
(121,249)(122,252)(123,251)(124,250)(125,245)(126,248)(127,247)(128,246)
(129,273)(130,276)(131,275)(132,274)(133,285)(134,288)(135,287)(136,286)
(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)(144,278);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 40)( 24, 39)
( 25, 42)( 26, 41)( 27, 44)( 28, 43)( 29, 46)( 30, 45)( 31, 48)( 32, 47)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 82)( 66, 81)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 88)( 72, 87)
( 73, 90)( 74, 89)( 75, 92)( 76, 91)( 77, 94)( 78, 93)( 79, 96)( 80, 95)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)(120,135)
(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)(128,143)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,178)(162,177)(163,180)(164,179)(165,182)(166,181)(167,184)(168,183)
(169,186)(170,185)(171,188)(172,187)(173,190)(174,189)(175,192)(176,191)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,226)(210,225)(211,228)(212,227)(213,230)(214,229)(215,232)(216,231)
(217,234)(218,233)(219,236)(220,235)(221,238)(222,237)(223,240)(224,239)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,274)(258,273)(259,276)(260,275)(261,278)(262,277)(263,280)(264,279)
(265,282)(266,281)(267,284)(268,283)(269,286)(270,285)(271,288)(272,287);;
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*s0*s1,
s0*s1*s2*s0*s1*s2*s0*s1*s2, s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 1,153)( 2,154)( 3,155)( 4,156)( 5,157)( 6,158)( 7,159)
( 8,160)( 9,145)( 10,146)( 11,147)( 12,148)( 13,149)( 14,150)( 15,151)
( 16,152)( 17,169)( 18,170)( 19,171)( 20,172)( 21,173)( 22,174)( 23,175)
( 24,176)( 25,161)( 26,162)( 27,163)( 28,164)( 29,165)( 30,166)( 31,167)
( 32,168)( 33,185)( 34,186)( 35,187)( 36,188)( 37,189)( 38,190)( 39,191)
( 40,192)( 41,177)( 42,178)( 43,179)( 44,180)( 45,181)( 46,182)( 47,183)
( 48,184)( 49,201)( 50,202)( 51,203)( 52,204)( 53,205)( 54,206)( 55,207)
( 56,208)( 57,193)( 58,194)( 59,195)( 60,196)( 61,197)( 62,198)( 63,199)
( 64,200)( 65,217)( 66,218)( 67,219)( 68,220)( 69,221)( 70,222)( 71,223)
( 72,224)( 73,209)( 74,210)( 75,211)( 76,212)( 77,213)( 78,214)( 79,215)
( 80,216)( 81,233)( 82,234)( 83,235)( 84,236)( 85,237)( 86,238)( 87,239)
( 88,240)( 89,225)( 90,226)( 91,227)( 92,228)( 93,229)( 94,230)( 95,231)
( 96,232)( 97,249)( 98,250)( 99,251)(100,252)(101,253)(102,254)(103,255)
(104,256)(105,241)(106,242)(107,243)(108,244)(109,245)(110,246)(111,247)
(112,248)(113,265)(114,266)(115,267)(116,268)(117,269)(118,270)(119,271)
(120,272)(121,257)(122,258)(123,259)(124,260)(125,261)(126,262)(127,263)
(128,264)(129,281)(130,282)(131,283)(132,284)(133,285)(134,286)(135,287)
(136,288)(137,273)(138,274)(139,275)(140,276)(141,277)(142,278)(143,279)
(144,280);
s1 := Sym(288)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49, 97)
( 50, 98)( 51,100)( 52, 99)( 53,101)( 54,102)( 55,104)( 56,103)( 57,109)
( 58,110)( 59,112)( 60,111)( 61,105)( 62,106)( 63,108)( 64,107)( 65,129)
( 66,130)( 67,132)( 68,131)( 69,133)( 70,134)( 71,136)( 72,135)( 73,141)
( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,113)
( 82,114)( 83,116)( 84,115)( 85,117)( 86,118)( 87,120)( 88,119)( 89,125)
( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)(147,148)
(151,152)(153,157)(154,158)(155,160)(156,159)(161,177)(162,178)(163,180)
(164,179)(165,181)(166,182)(167,184)(168,183)(169,189)(170,190)(171,192)
(172,191)(173,185)(174,186)(175,188)(176,187)(193,241)(194,242)(195,244)
(196,243)(197,245)(198,246)(199,248)(200,247)(201,253)(202,254)(203,256)
(204,255)(205,249)(206,250)(207,252)(208,251)(209,273)(210,274)(211,276)
(212,275)(213,277)(214,278)(215,280)(216,279)(217,285)(218,286)(219,288)
(220,287)(221,281)(222,282)(223,284)(224,283)(225,257)(226,258)(227,260)
(228,259)(229,261)(230,262)(231,264)(232,263)(233,269)(234,270)(235,272)
(236,271)(237,265)(238,266)(239,268)(240,267);
s2 := Sym(288)!( 1,209)( 2,212)( 3,211)( 4,210)( 5,221)( 6,224)( 7,223)
( 8,222)( 9,217)( 10,220)( 11,219)( 12,218)( 13,213)( 14,216)( 15,215)
( 16,214)( 17,193)( 18,196)( 19,195)( 20,194)( 21,205)( 22,208)( 23,207)
( 24,206)( 25,201)( 26,204)( 27,203)( 28,202)( 29,197)( 30,200)( 31,199)
( 32,198)( 33,225)( 34,228)( 35,227)( 36,226)( 37,237)( 38,240)( 39,239)
( 40,238)( 41,233)( 42,236)( 43,235)( 44,234)( 45,229)( 46,232)( 47,231)
( 48,230)( 49,161)( 50,164)( 51,163)( 52,162)( 53,173)( 54,176)( 55,175)
( 56,174)( 57,169)( 58,172)( 59,171)( 60,170)( 61,165)( 62,168)( 63,167)
( 64,166)( 65,145)( 66,148)( 67,147)( 68,146)( 69,157)( 70,160)( 71,159)
( 72,158)( 73,153)( 74,156)( 75,155)( 76,154)( 77,149)( 78,152)( 79,151)
( 80,150)( 81,177)( 82,180)( 83,179)( 84,178)( 85,189)( 86,192)( 87,191)
( 88,190)( 89,185)( 90,188)( 91,187)( 92,186)( 93,181)( 94,184)( 95,183)
( 96,182)( 97,257)( 98,260)( 99,259)(100,258)(101,269)(102,272)(103,271)
(104,270)(105,265)(106,268)(107,267)(108,266)(109,261)(110,264)(111,263)
(112,262)(113,241)(114,244)(115,243)(116,242)(117,253)(118,256)(119,255)
(120,254)(121,249)(122,252)(123,251)(124,250)(125,245)(126,248)(127,247)
(128,246)(129,273)(130,276)(131,275)(132,274)(133,285)(134,288)(135,287)
(136,286)(137,281)(138,284)(139,283)(140,282)(141,277)(142,280)(143,279)
(144,278);
s3 := Sym(288)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 34)( 18, 33)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 40)
( 24, 39)( 25, 42)( 26, 41)( 27, 44)( 28, 43)( 29, 46)( 30, 45)( 31, 48)
( 32, 47)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 82)( 66, 81)( 67, 84)( 68, 83)( 69, 86)( 70, 85)( 71, 88)
( 72, 87)( 73, 90)( 74, 89)( 75, 92)( 76, 91)( 77, 94)( 78, 93)( 79, 96)
( 80, 95)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,130)(114,129)(115,132)(116,131)(117,134)(118,133)(119,136)
(120,135)(121,138)(122,137)(123,140)(124,139)(125,142)(126,141)(127,144)
(128,143)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,178)(162,177)(163,180)(164,179)(165,182)(166,181)(167,184)
(168,183)(169,186)(170,185)(171,188)(172,187)(173,190)(174,189)(175,192)
(176,191)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,226)(210,225)(211,228)(212,227)(213,230)(214,229)(215,232)
(216,231)(217,234)(218,233)(219,236)(220,235)(221,238)(222,237)(223,240)
(224,239)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,274)(258,273)(259,276)(260,275)(261,278)(262,277)(263,280)
(264,279)(265,282)(266,281)(267,284)(268,283)(269,286)(270,285)(271,288)
(272,287);
poly := sub<Sym(288)|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*s0*s1, s0*s1*s2*s0*s1*s2*s0*s1*s2,
s0*s1*s2*s3*s2*s1*s0*s1*s2*s3*s2*s1,
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2,
s3*s1*s2*s3*s2*s3*s2*s3*s2*s3*s1*s2*s3*s2*s3*s2*s3*s2 >;
References : None.
to this polytope