Polytope of Type {4,4,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,9}*1152a
if this polytope has a name.
Group : SmallGroup(1152,153992)
Rank : 4
Schlafli Type : {4,4,9}
Number of vertices, edges, etc : 16, 32, 72, 9
Order of s0s1s2s3 : 18
Order of s0s1s2s3s2s1 : 4
Special Properties :
Universal
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 : {4,4,9}*576a
3-fold quotients : {4,4,3}*384a
6-fold quotients : {4,4,3}*192a
8-fold quotients : {2,4,9}*144
24-fold quotients : {2,4,3}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 22)( 18, 21)( 19, 24)( 20, 23)
( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 41, 44)( 42, 43)( 45, 48)( 46, 47)
( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 64)( 58, 63)( 59, 62)( 60, 61)
( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 86)( 82, 85)( 83, 88)( 84, 87)
( 89, 96)( 90, 95)( 91, 94)( 92, 93)(105,108)(106,107)(109,112)(110,111)
(113,118)(114,117)(115,120)(116,119)(121,128)(122,127)(123,126)(124,125)
(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)(148,151)
(153,160)(154,159)(155,158)(156,157)(169,172)(170,171)(173,176)(174,175)
(177,182)(178,181)(179,184)(180,183)(185,192)(186,191)(187,190)(188,189)
(201,204)(202,203)(205,208)(206,207)(209,214)(210,213)(211,216)(212,215)
(217,224)(218,223)(219,222)(220,221)(233,236)(234,235)(237,240)(238,239)
(241,246)(242,245)(243,248)(244,247)(249,256)(250,255)(251,254)(252,253)
(265,268)(266,267)(269,272)(270,271)(273,278)(274,277)(275,280)(276,279)
(281,288)(282,287)(283,286)(284,285);;
s1 := ( 1, 17)( 2, 18)( 3, 20)( 4, 19)( 5, 21)( 6, 22)( 7, 24)( 8, 23)
( 9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 29)( 14, 30)( 15, 32)( 16, 31)
( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 53)( 38, 54)( 39, 56)( 40, 55)
( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 61)( 46, 62)( 47, 64)( 48, 63)
( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)( 72, 87)
( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 93)( 78, 94)( 79, 96)( 80, 95)
( 97,113)( 98,114)( 99,116)(100,115)(101,117)(102,118)(103,120)(104,119)
(105,121)(106,122)(107,124)(108,123)(109,125)(110,126)(111,128)(112,127)
(129,145)(130,146)(131,148)(132,147)(133,149)(134,150)(135,152)(136,151)
(137,153)(138,154)(139,156)(140,155)(141,157)(142,158)(143,160)(144,159)
(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)(168,183)
(169,185)(170,186)(171,188)(172,187)(173,189)(174,190)(175,192)(176,191)
(193,209)(194,210)(195,212)(196,211)(197,213)(198,214)(199,216)(200,215)
(201,217)(202,218)(203,220)(204,219)(205,221)(206,222)(207,224)(208,223)
(225,241)(226,242)(227,244)(228,243)(229,245)(230,246)(231,248)(232,247)
(233,249)(234,250)(235,252)(236,251)(237,253)(238,254)(239,256)(240,255)
(257,273)(258,274)(259,276)(260,275)(261,277)(262,278)(263,280)(264,279)
(265,281)(266,282)(267,284)(268,283)(269,285)(270,286)(271,288)(272,287);;
s2 := ( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)
( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)
( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)
( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)
( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)
( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97,257)( 98,258)( 99,259)(100,260)
(101,263)(102,264)(103,261)(104,262)(105,265)(106,266)(107,267)(108,268)
(109,271)(110,272)(111,269)(112,270)(113,281)(114,282)(115,283)(116,284)
(117,287)(118,288)(119,285)(120,286)(121,273)(122,274)(123,275)(124,276)
(125,279)(126,280)(127,277)(128,278)(129,225)(130,226)(131,227)(132,228)
(133,231)(134,232)(135,229)(136,230)(137,233)(138,234)(139,235)(140,236)
(141,239)(142,240)(143,237)(144,238)(145,249)(146,250)(147,251)(148,252)
(149,255)(150,256)(151,253)(152,254)(153,241)(154,242)(155,243)(156,244)
(157,247)(158,248)(159,245)(160,246)(161,193)(162,194)(163,195)(164,196)
(165,199)(166,200)(167,197)(168,198)(169,201)(170,202)(171,203)(172,204)
(173,207)(174,208)(175,205)(176,206)(177,217)(178,218)(179,219)(180,220)
(181,223)(182,224)(183,221)(184,222)(185,209)(186,210)(187,211)(188,212)
(189,215)(190,216)(191,213)(192,214);;
s3 := ( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,101)( 6,102)( 7, 99)( 8,100)
( 9,121)( 10,122)( 11,127)( 12,128)( 13,125)( 14,126)( 15,123)( 16,124)
( 17,113)( 18,114)( 19,119)( 20,120)( 21,117)( 22,118)( 23,115)( 24,116)
( 25,105)( 26,106)( 27,111)( 28,112)( 29,109)( 30,110)( 31,107)( 32,108)
( 33,161)( 34,162)( 35,167)( 36,168)( 37,165)( 38,166)( 39,163)( 40,164)
( 41,185)( 42,186)( 43,191)( 44,192)( 45,189)( 46,190)( 47,187)( 48,188)
( 49,177)( 50,178)( 51,183)( 52,184)( 53,181)( 54,182)( 55,179)( 56,180)
( 57,169)( 58,170)( 59,175)( 60,176)( 61,173)( 62,174)( 63,171)( 64,172)
( 65,129)( 66,130)( 67,135)( 68,136)( 69,133)( 70,134)( 71,131)( 72,132)
( 73,153)( 74,154)( 75,159)( 76,160)( 77,157)( 78,158)( 79,155)( 80,156)
( 81,145)( 82,146)( 83,151)( 84,152)( 85,149)( 86,150)( 87,147)( 88,148)
( 89,137)( 90,138)( 91,143)( 92,144)( 93,141)( 94,142)( 95,139)( 96,140)
(193,257)(194,258)(195,263)(196,264)(197,261)(198,262)(199,259)(200,260)
(201,281)(202,282)(203,287)(204,288)(205,285)(206,286)(207,283)(208,284)
(209,273)(210,274)(211,279)(212,280)(213,277)(214,278)(215,275)(216,276)
(217,265)(218,266)(219,271)(220,272)(221,269)(222,270)(223,267)(224,268)
(227,231)(228,232)(233,249)(234,250)(235,255)(236,256)(237,253)(238,254)
(239,251)(240,252)(243,247)(244,248);;
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,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(288)!( 9, 12)( 10, 11)( 13, 16)( 14, 15)( 17, 22)( 18, 21)( 19, 24)
( 20, 23)( 25, 32)( 26, 31)( 27, 30)( 28, 29)( 41, 44)( 42, 43)( 45, 48)
( 46, 47)( 49, 54)( 50, 53)( 51, 56)( 52, 55)( 57, 64)( 58, 63)( 59, 62)
( 60, 61)( 73, 76)( 74, 75)( 77, 80)( 78, 79)( 81, 86)( 82, 85)( 83, 88)
( 84, 87)( 89, 96)( 90, 95)( 91, 94)( 92, 93)(105,108)(106,107)(109,112)
(110,111)(113,118)(114,117)(115,120)(116,119)(121,128)(122,127)(123,126)
(124,125)(137,140)(138,139)(141,144)(142,143)(145,150)(146,149)(147,152)
(148,151)(153,160)(154,159)(155,158)(156,157)(169,172)(170,171)(173,176)
(174,175)(177,182)(178,181)(179,184)(180,183)(185,192)(186,191)(187,190)
(188,189)(201,204)(202,203)(205,208)(206,207)(209,214)(210,213)(211,216)
(212,215)(217,224)(218,223)(219,222)(220,221)(233,236)(234,235)(237,240)
(238,239)(241,246)(242,245)(243,248)(244,247)(249,256)(250,255)(251,254)
(252,253)(265,268)(266,267)(269,272)(270,271)(273,278)(274,277)(275,280)
(276,279)(281,288)(282,287)(283,286)(284,285);
s1 := Sym(288)!( 1, 17)( 2, 18)( 3, 20)( 4, 19)( 5, 21)( 6, 22)( 7, 24)
( 8, 23)( 9, 25)( 10, 26)( 11, 28)( 12, 27)( 13, 29)( 14, 30)( 15, 32)
( 16, 31)( 33, 49)( 34, 50)( 35, 52)( 36, 51)( 37, 53)( 38, 54)( 39, 56)
( 40, 55)( 41, 57)( 42, 58)( 43, 60)( 44, 59)( 45, 61)( 46, 62)( 47, 64)
( 48, 63)( 65, 81)( 66, 82)( 67, 84)( 68, 83)( 69, 85)( 70, 86)( 71, 88)
( 72, 87)( 73, 89)( 74, 90)( 75, 92)( 76, 91)( 77, 93)( 78, 94)( 79, 96)
( 80, 95)( 97,113)( 98,114)( 99,116)(100,115)(101,117)(102,118)(103,120)
(104,119)(105,121)(106,122)(107,124)(108,123)(109,125)(110,126)(111,128)
(112,127)(129,145)(130,146)(131,148)(132,147)(133,149)(134,150)(135,152)
(136,151)(137,153)(138,154)(139,156)(140,155)(141,157)(142,158)(143,160)
(144,159)(161,177)(162,178)(163,180)(164,179)(165,181)(166,182)(167,184)
(168,183)(169,185)(170,186)(171,188)(172,187)(173,189)(174,190)(175,192)
(176,191)(193,209)(194,210)(195,212)(196,211)(197,213)(198,214)(199,216)
(200,215)(201,217)(202,218)(203,220)(204,219)(205,221)(206,222)(207,224)
(208,223)(225,241)(226,242)(227,244)(228,243)(229,245)(230,246)(231,248)
(232,247)(233,249)(234,250)(235,252)(236,251)(237,253)(238,254)(239,256)
(240,255)(257,273)(258,274)(259,276)(260,275)(261,277)(262,278)(263,280)
(264,279)(265,281)(266,282)(267,284)(268,283)(269,285)(270,286)(271,288)
(272,287);
s2 := Sym(288)!( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)
( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)
( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)
( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)
( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)
( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)( 97,257)( 98,258)( 99,259)
(100,260)(101,263)(102,264)(103,261)(104,262)(105,265)(106,266)(107,267)
(108,268)(109,271)(110,272)(111,269)(112,270)(113,281)(114,282)(115,283)
(116,284)(117,287)(118,288)(119,285)(120,286)(121,273)(122,274)(123,275)
(124,276)(125,279)(126,280)(127,277)(128,278)(129,225)(130,226)(131,227)
(132,228)(133,231)(134,232)(135,229)(136,230)(137,233)(138,234)(139,235)
(140,236)(141,239)(142,240)(143,237)(144,238)(145,249)(146,250)(147,251)
(148,252)(149,255)(150,256)(151,253)(152,254)(153,241)(154,242)(155,243)
(156,244)(157,247)(158,248)(159,245)(160,246)(161,193)(162,194)(163,195)
(164,196)(165,199)(166,200)(167,197)(168,198)(169,201)(170,202)(171,203)
(172,204)(173,207)(174,208)(175,205)(176,206)(177,217)(178,218)(179,219)
(180,220)(181,223)(182,224)(183,221)(184,222)(185,209)(186,210)(187,211)
(188,212)(189,215)(190,216)(191,213)(192,214);
s3 := Sym(288)!( 1, 97)( 2, 98)( 3,103)( 4,104)( 5,101)( 6,102)( 7, 99)
( 8,100)( 9,121)( 10,122)( 11,127)( 12,128)( 13,125)( 14,126)( 15,123)
( 16,124)( 17,113)( 18,114)( 19,119)( 20,120)( 21,117)( 22,118)( 23,115)
( 24,116)( 25,105)( 26,106)( 27,111)( 28,112)( 29,109)( 30,110)( 31,107)
( 32,108)( 33,161)( 34,162)( 35,167)( 36,168)( 37,165)( 38,166)( 39,163)
( 40,164)( 41,185)( 42,186)( 43,191)( 44,192)( 45,189)( 46,190)( 47,187)
( 48,188)( 49,177)( 50,178)( 51,183)( 52,184)( 53,181)( 54,182)( 55,179)
( 56,180)( 57,169)( 58,170)( 59,175)( 60,176)( 61,173)( 62,174)( 63,171)
( 64,172)( 65,129)( 66,130)( 67,135)( 68,136)( 69,133)( 70,134)( 71,131)
( 72,132)( 73,153)( 74,154)( 75,159)( 76,160)( 77,157)( 78,158)( 79,155)
( 80,156)( 81,145)( 82,146)( 83,151)( 84,152)( 85,149)( 86,150)( 87,147)
( 88,148)( 89,137)( 90,138)( 91,143)( 92,144)( 93,141)( 94,142)( 95,139)
( 96,140)(193,257)(194,258)(195,263)(196,264)(197,261)(198,262)(199,259)
(200,260)(201,281)(202,282)(203,287)(204,288)(205,285)(206,286)(207,283)
(208,284)(209,273)(210,274)(211,279)(212,280)(213,277)(214,278)(215,275)
(216,276)(217,265)(218,266)(219,271)(220,272)(221,269)(222,270)(223,267)
(224,268)(227,231)(228,232)(233,249)(234,250)(235,255)(236,256)(237,253)
(238,254)(239,251)(240,252)(243,247)(244,248);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope