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# Polytope of Type {72,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {72,4}*1152d
if this polytope has a name.
Group : SmallGroup(1152,154353)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 144, 288, 8
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,4}*576b
3-fold quotients : {24,4}*384d
4-fold quotients : {36,4}*288b, {36,4}*288c, {18,4}*288
6-fold quotients : {12,4}*192b
8-fold quotients : {36,2}*144, {9,4}*144, {18,4}*144b, {18,4}*144c
12-fold quotients : {12,4}*96b, {12,4}*96c, {6,4}*96
16-fold quotients : {9,4}*72, {18,2}*72
24-fold quotients : {12,2}*48, {3,4}*48, {6,4}*48b, {6,4}*48c
32-fold quotients : {9,2}*36
48-fold quotients : {3,4}*24, {6,2}*24
72-fold quotients : {4,2}*16
96-fold quotients : {3,2}*12
144-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)( 50, 70)
( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)( 58, 62)
( 59, 64)( 60, 63)( 73,109)( 74,110)( 75,112)( 76,111)( 77,117)( 78,118)
( 79,120)( 80,119)( 81,113)( 82,114)( 83,116)( 84,115)( 85,141)( 86,142)
( 87,144)( 88,143)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)( 94,134)
( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,125)(102,126)
(103,128)(104,127)(105,121)(106,122)(107,124)(108,123)(145,217)(146,218)
(147,220)(148,219)(149,225)(150,226)(151,228)(152,227)(153,221)(154,222)
(155,224)(156,223)(157,249)(158,250)(159,252)(160,251)(161,245)(162,246)
(163,248)(164,247)(165,241)(166,242)(167,244)(168,243)(169,237)(170,238)
(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)(177,229)(178,230)
(179,232)(180,231)(181,253)(182,254)(183,256)(184,255)(185,261)(186,262)
(187,264)(188,263)(189,257)(190,258)(191,260)(192,259)(193,285)(194,286)
(195,288)(196,287)(197,281)(198,282)(199,284)(200,283)(201,277)(202,278)
(203,280)(204,279)(205,273)(206,274)(207,276)(208,275)(209,269)(210,270)
(211,272)(212,271)(213,265)(214,266)(215,268)(216,267);;
s1 := (  1,157)(  2,160)(  3,159)(  4,158)(  5,165)(  6,168)(  7,167)(  8,166)
(  9,161)( 10,164)( 11,163)( 12,162)( 13,145)( 14,148)( 15,147)( 16,146)
( 17,153)( 18,156)( 19,155)( 20,154)( 21,149)( 22,152)( 23,151)( 24,150)
( 25,177)( 26,180)( 27,179)( 28,178)( 29,173)( 30,176)( 31,175)( 32,174)
( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)( 40,194)
( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)( 48,198)
( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)( 56,190)
( 57,185)( 58,188)( 59,187)( 60,186)( 61,213)( 62,216)( 63,215)( 64,214)
( 65,209)( 66,212)( 67,211)( 68,210)( 69,205)( 70,208)( 71,207)( 72,206)
( 73,265)( 74,268)( 75,267)( 76,266)( 77,273)( 78,276)( 79,275)( 80,274)
( 81,269)( 82,272)( 83,271)( 84,270)( 85,253)( 86,256)( 87,255)( 88,254)
( 89,261)( 90,264)( 91,263)( 92,262)( 93,257)( 94,260)( 95,259)( 96,258)
( 97,285)( 98,288)( 99,287)(100,286)(101,281)(102,284)(103,283)(104,282)
(105,277)(106,280)(107,279)(108,278)(109,229)(110,232)(111,231)(112,230)
(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)(120,234)
(121,217)(122,220)(123,219)(124,218)(125,225)(126,228)(127,227)(128,226)
(129,221)(130,224)(131,223)(132,222)(133,249)(134,252)(135,251)(136,250)
(137,245)(138,248)(139,247)(140,246)(141,241)(142,244)(143,243)(144,242);;
s2 := (  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)(152,187)
(153,190)(154,189)(155,192)(156,191)(157,194)(158,193)(159,196)(160,195)
(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)(168,203)
(169,206)(170,205)(171,208)(172,207)(173,210)(174,209)(175,212)(176,211)
(177,214)(178,213)(179,216)(180,215)(217,254)(218,253)(219,256)(220,255)
(221,258)(222,257)(223,260)(224,259)(225,262)(226,261)(227,264)(228,263)
(229,266)(230,265)(231,268)(232,267)(233,270)(234,269)(235,272)(236,271)
(237,274)(238,273)(239,276)(240,275)(241,278)(242,277)(243,280)(244,279)
(245,282)(246,281)(247,284)(248,283)(249,286)(250,285)(251,288)(252,287);;
poly := Group([s0,s1,s2]);;

```
Finitely Presented Group Representation (GAP) :
```F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*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*s2*s1*s0*s1*s0*s1*s0*s2*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(288)!(  3,  4)(  5,  9)(  6, 10)(  7, 12)(  8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 49, 69)
( 50, 70)( 51, 72)( 52, 71)( 53, 65)( 54, 66)( 55, 68)( 56, 67)( 57, 61)
( 58, 62)( 59, 64)( 60, 63)( 73,109)( 74,110)( 75,112)( 76,111)( 77,117)
( 78,118)( 79,120)( 80,119)( 81,113)( 82,114)( 83,116)( 84,115)( 85,141)
( 86,142)( 87,144)( 88,143)( 89,137)( 90,138)( 91,140)( 92,139)( 93,133)
( 94,134)( 95,136)( 96,135)( 97,129)( 98,130)( 99,132)(100,131)(101,125)
(102,126)(103,128)(104,127)(105,121)(106,122)(107,124)(108,123)(145,217)
(146,218)(147,220)(148,219)(149,225)(150,226)(151,228)(152,227)(153,221)
(154,222)(155,224)(156,223)(157,249)(158,250)(159,252)(160,251)(161,245)
(162,246)(163,248)(164,247)(165,241)(166,242)(167,244)(168,243)(169,237)
(170,238)(171,240)(172,239)(173,233)(174,234)(175,236)(176,235)(177,229)
(178,230)(179,232)(180,231)(181,253)(182,254)(183,256)(184,255)(185,261)
(186,262)(187,264)(188,263)(189,257)(190,258)(191,260)(192,259)(193,285)
(194,286)(195,288)(196,287)(197,281)(198,282)(199,284)(200,283)(201,277)
(202,278)(203,280)(204,279)(205,273)(206,274)(207,276)(208,275)(209,269)
(210,270)(211,272)(212,271)(213,265)(214,266)(215,268)(216,267);
s1 := Sym(288)!(  1,157)(  2,160)(  3,159)(  4,158)(  5,165)(  6,168)(  7,167)
(  8,166)(  9,161)( 10,164)( 11,163)( 12,162)( 13,145)( 14,148)( 15,147)
( 16,146)( 17,153)( 18,156)( 19,155)( 20,154)( 21,149)( 22,152)( 23,151)
( 24,150)( 25,177)( 26,180)( 27,179)( 28,178)( 29,173)( 30,176)( 31,175)
( 32,174)( 33,169)( 34,172)( 35,171)( 36,170)( 37,193)( 38,196)( 39,195)
( 40,194)( 41,201)( 42,204)( 43,203)( 44,202)( 45,197)( 46,200)( 47,199)
( 48,198)( 49,181)( 50,184)( 51,183)( 52,182)( 53,189)( 54,192)( 55,191)
( 56,190)( 57,185)( 58,188)( 59,187)( 60,186)( 61,213)( 62,216)( 63,215)
( 64,214)( 65,209)( 66,212)( 67,211)( 68,210)( 69,205)( 70,208)( 71,207)
( 72,206)( 73,265)( 74,268)( 75,267)( 76,266)( 77,273)( 78,276)( 79,275)
( 80,274)( 81,269)( 82,272)( 83,271)( 84,270)( 85,253)( 86,256)( 87,255)
( 88,254)( 89,261)( 90,264)( 91,263)( 92,262)( 93,257)( 94,260)( 95,259)
( 96,258)( 97,285)( 98,288)( 99,287)(100,286)(101,281)(102,284)(103,283)
(104,282)(105,277)(106,280)(107,279)(108,278)(109,229)(110,232)(111,231)
(112,230)(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)
(120,234)(121,217)(122,220)(123,219)(124,218)(125,225)(126,228)(127,227)
(128,226)(129,221)(130,224)(131,223)(132,222)(133,249)(134,252)(135,251)
(136,250)(137,245)(138,248)(139,247)(140,246)(141,241)(142,244)(143,243)
(144,242);
s2 := Sym(288)!(  1,  2)(  3,  4)(  5,  6)(  7,  8)(  9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,182)(146,181)(147,184)(148,183)(149,186)(150,185)(151,188)
(152,187)(153,190)(154,189)(155,192)(156,191)(157,194)(158,193)(159,196)
(160,195)(161,198)(162,197)(163,200)(164,199)(165,202)(166,201)(167,204)
(168,203)(169,206)(170,205)(171,208)(172,207)(173,210)(174,209)(175,212)
(176,211)(177,214)(178,213)(179,216)(180,215)(217,254)(218,253)(219,256)
(220,255)(221,258)(222,257)(223,260)(224,259)(225,262)(226,261)(227,264)
(228,263)(229,266)(230,265)(231,268)(232,267)(233,270)(234,269)(235,272)
(236,271)(237,274)(238,273)(239,276)(240,275)(241,278)(242,277)(243,280)
(244,279)(245,282)(246,281)(247,284)(248,283)(249,286)(250,285)(251,288)
(252,287);
poly := sub<Sym(288)|s0,s1,s2>;

```
Finitely Presented Group Representation (Magma) :
```poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2,
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s2*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*s2*s1*s0*s1*s0*s1*s0*s2*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