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# Polytope of Type {72,4}

Atlas Canonical Name : {72,4}*576d
if this polytope has a name.
Group : SmallGroup(576,1386)
Rank : 3
Schlafli Type : {72,4}
Number of vertices, edges, etc : 72, 144, 4
Order of s0s1s2 : 72
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{72,4,2} of size 1152
Vertex Figure Of :
{2,72,4} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {36,4}*288b
3-fold quotients : {24,4}*192d
4-fold quotients : {18,4}*144b
6-fold quotients : {12,4}*96b
8-fold quotients : {9,4}*72
12-fold quotients : {6,4}*48c
24-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
2-fold covers : {72,4}*1152c
3-fold covers : {216,4}*1728d
Permutation Representation (GAP) :
```s0 := (  2,  3)(  5,  9)(  6, 11)(  7, 10)(  8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 38, 39)( 41, 45)( 42, 47)( 43, 46)( 44, 48)( 49, 65)( 50, 67)
( 51, 66)( 52, 68)( 53, 61)( 54, 63)( 55, 62)( 56, 64)( 57, 69)( 58, 71)
( 59, 70)( 60, 72)( 73,109)( 74,111)( 75,110)( 76,112)( 77,117)( 78,119)
( 79,118)( 80,120)( 81,113)( 82,115)( 83,114)( 84,116)( 85,137)( 86,139)
( 87,138)( 88,140)( 89,133)( 90,135)( 91,134)( 92,136)( 93,141)( 94,143)
( 95,142)( 96,144)( 97,125)( 98,127)( 99,126)(100,128)(101,121)(102,123)
(103,122)(104,124)(105,129)(106,131)(107,130)(108,132)(145,217)(146,219)
(147,218)(148,220)(149,225)(150,227)(151,226)(152,228)(153,221)(154,223)
(155,222)(156,224)(157,245)(158,247)(159,246)(160,248)(161,241)(162,243)
(163,242)(164,244)(165,249)(166,251)(167,250)(168,252)(169,233)(170,235)
(171,234)(172,236)(173,229)(174,231)(175,230)(176,232)(177,237)(178,239)
(179,238)(180,240)(181,253)(182,255)(183,254)(184,256)(185,261)(186,263)
(187,262)(188,264)(189,257)(190,259)(191,258)(192,260)(193,281)(194,283)
(195,282)(196,284)(197,277)(198,279)(199,278)(200,280)(201,285)(202,287)
(203,286)(204,288)(205,269)(206,271)(207,270)(208,272)(209,265)(210,267)
(211,266)(212,268)(213,273)(214,275)(215,274)(216,276);;
s1 := (  1,157)(  2,158)(  3,160)(  4,159)(  5,165)(  6,166)(  7,168)(  8,167)
(  9,161)( 10,162)( 11,164)( 12,163)( 13,145)( 14,146)( 15,148)( 16,147)
( 17,153)( 18,154)( 19,156)( 20,155)( 21,149)( 22,150)( 23,152)( 24,151)
( 25,173)( 26,174)( 27,176)( 28,175)( 29,169)( 30,170)( 31,172)( 32,171)
( 33,177)( 34,178)( 35,180)( 36,179)( 37,193)( 38,194)( 39,196)( 40,195)
( 41,201)( 42,202)( 43,204)( 44,203)( 45,197)( 46,198)( 47,200)( 48,199)
( 49,181)( 50,182)( 51,184)( 52,183)( 53,189)( 54,190)( 55,192)( 56,191)
( 57,185)( 58,186)( 59,188)( 60,187)( 61,209)( 62,210)( 63,212)( 64,211)
( 65,205)( 66,206)( 67,208)( 68,207)( 69,213)( 70,214)( 71,216)( 72,215)
( 73,265)( 74,266)( 75,268)( 76,267)( 77,273)( 78,274)( 79,276)( 80,275)
( 81,269)( 82,270)( 83,272)( 84,271)( 85,253)( 86,254)( 87,256)( 88,255)
( 89,261)( 90,262)( 91,264)( 92,263)( 93,257)( 94,258)( 95,260)( 96,259)
( 97,281)( 98,282)( 99,284)(100,283)(101,277)(102,278)(103,280)(104,279)
(105,285)(106,286)(107,288)(108,287)(109,229)(110,230)(111,232)(112,231)
(113,237)(114,238)(115,240)(116,239)(117,233)(118,234)(119,236)(120,235)
(121,217)(122,218)(123,220)(124,219)(125,225)(126,226)(127,228)(128,227)
(129,221)(130,222)(131,224)(132,223)(133,245)(134,246)(135,248)(136,247)
(137,241)(138,242)(139,244)(140,243)(141,249)(142,250)(143,252)(144,251);;
s2 := (  1, 40)(  2, 39)(  3, 38)(  4, 37)(  5, 44)(  6, 43)(  7, 42)(  8, 41)
(  9, 48)( 10, 47)( 11, 46)( 12, 45)( 13, 52)( 14, 51)( 15, 50)( 16, 49)
( 17, 56)( 18, 55)( 19, 54)( 20, 53)( 21, 60)( 22, 59)( 23, 58)( 24, 57)
( 25, 64)( 26, 63)( 27, 62)( 28, 61)( 29, 68)( 30, 67)( 31, 66)( 32, 65)
( 33, 72)( 34, 71)( 35, 70)( 36, 69)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,116)( 78,115)( 79,114)( 80,113)( 81,120)( 82,119)( 83,118)( 84,117)
( 85,124)( 86,123)( 87,122)( 88,121)( 89,128)( 90,127)( 91,126)( 92,125)
( 93,132)( 94,131)( 95,130)( 96,129)( 97,136)( 98,135)( 99,134)(100,133)
(101,140)(102,139)(103,138)(104,137)(105,144)(106,143)(107,142)(108,141)
(145,184)(146,183)(147,182)(148,181)(149,188)(150,187)(151,186)(152,185)
(153,192)(154,191)(155,190)(156,189)(157,196)(158,195)(159,194)(160,193)
(161,200)(162,199)(163,198)(164,197)(165,204)(166,203)(167,202)(168,201)
(169,208)(170,207)(171,206)(172,205)(173,212)(174,211)(175,210)(176,209)
(177,216)(178,215)(179,214)(180,213)(217,256)(218,255)(219,254)(220,253)
(221,260)(222,259)(223,258)(224,257)(225,264)(226,263)(227,262)(228,261)
(229,268)(230,267)(231,266)(232,265)(233,272)(234,271)(235,270)(236,269)
(237,276)(238,275)(239,274)(240,273)(241,280)(242,279)(243,278)(244,277)
(245,284)(246,283)(247,282)(248,281)(249,288)(250,287)(251,286)(252,285);;
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*s0*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*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*s2*s0*s1*s2*s0*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope