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# Polytope of Type {16,6}

Atlas Canonical Name : {16,6}*576
if this polytope has a name.
Group : SmallGroup(576,5158)
Rank : 3
Schlafli Type : {16,6}
Number of vertices, edges, etc : 48, 144, 18
Order of s0s1s2 : 16
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
{16,6,2} of size 1152
{16,6,3} of size 1728
Vertex Figure Of :
{2,16,6} of size 1152
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {8,6}*288
4-fold quotients : {4,6}*144
8-fold quotients : {4,6}*72
9-fold quotients : {16,2}*64
18-fold quotients : {8,2}*32
36-fold quotients : {4,2}*16
72-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {16,12}*1152a, {32,6}*1152
3-fold covers : {16,6}*1728a, {48,6}*1728d, {48,6}*1728e, {16,6}*1728b, {48,6}*1728g, {48,6}*1728h
Permutation Representation (GAP) :
```s0 := (  4,  7)(  5,  8)(  6,  9)( 13, 16)( 14, 17)( 15, 18)( 19, 28)( 20, 29)
( 21, 30)( 22, 34)( 23, 35)( 24, 36)( 25, 31)( 26, 32)( 27, 33)( 37, 55)
( 38, 56)( 39, 57)( 40, 61)( 41, 62)( 42, 63)( 43, 58)( 44, 59)( 45, 60)
( 46, 64)( 47, 65)( 48, 66)( 49, 70)( 50, 71)( 51, 72)( 52, 67)( 53, 68)
( 54, 69)( 73, 91)( 74, 92)( 75, 93)( 76, 97)( 77, 98)( 78, 99)( 79, 94)
( 80, 95)( 81, 96)( 82,100)( 83,101)( 84,102)( 85,106)( 86,107)( 87,108)
( 88,103)( 89,104)( 90,105)(109,118)(110,119)(111,120)(112,124)(113,125)
(114,126)(115,121)(116,122)(117,123)(130,133)(131,134)(132,135)(139,142)
(140,143)(141,144)(145,181)(146,182)(147,183)(148,187)(149,188)(150,189)
(151,184)(152,185)(153,186)(154,190)(155,191)(156,192)(157,196)(158,197)
(159,198)(160,193)(161,194)(162,195)(163,208)(164,209)(165,210)(166,214)
(167,215)(168,216)(169,211)(170,212)(171,213)(172,199)(173,200)(174,201)
(175,205)(176,206)(177,207)(178,202)(179,203)(180,204)(217,271)(218,272)
(219,273)(220,277)(221,278)(222,279)(223,274)(224,275)(225,276)(226,280)
(227,281)(228,282)(229,286)(230,287)(231,288)(232,283)(233,284)(234,285)
(235,253)(236,254)(237,255)(238,259)(239,260)(240,261)(241,256)(242,257)
(243,258)(244,262)(245,263)(246,264)(247,268)(248,269)(249,270)(250,265)
(251,266)(252,267);;
s1 := (  1,145)(  2,148)(  3,151)(  4,146)(  5,149)(  6,152)(  7,147)(  8,150)
(  9,153)( 10,154)( 11,157)( 12,160)( 13,155)( 14,158)( 15,161)( 16,156)
( 17,159)( 18,162)( 19,172)( 20,175)( 21,178)( 22,173)( 23,176)( 24,179)
( 25,174)( 26,177)( 27,180)( 28,163)( 29,166)( 30,169)( 31,164)( 32,167)
( 33,170)( 34,165)( 35,168)( 36,171)( 37,199)( 38,202)( 39,205)( 40,200)
( 41,203)( 42,206)( 43,201)( 44,204)( 45,207)( 46,208)( 47,211)( 48,214)
( 49,209)( 50,212)( 51,215)( 52,210)( 53,213)( 54,216)( 55,181)( 56,184)
( 57,187)( 58,182)( 59,185)( 60,188)( 61,183)( 62,186)( 63,189)( 64,190)
( 65,193)( 66,196)( 67,191)( 68,194)( 69,197)( 70,192)( 71,195)( 72,198)
( 73,235)( 74,238)( 75,241)( 76,236)( 77,239)( 78,242)( 79,237)( 80,240)
( 81,243)( 82,244)( 83,247)( 84,250)( 85,245)( 86,248)( 87,251)( 88,246)
( 89,249)( 90,252)( 91,217)( 92,220)( 93,223)( 94,218)( 95,221)( 96,224)
( 97,219)( 98,222)( 99,225)(100,226)(101,229)(102,232)(103,227)(104,230)
(105,233)(106,228)(107,231)(108,234)(109,262)(110,265)(111,268)(112,263)
(113,266)(114,269)(115,264)(116,267)(117,270)(118,253)(119,256)(120,259)
(121,254)(122,257)(123,260)(124,255)(125,258)(126,261)(127,271)(128,274)
(129,277)(130,272)(131,275)(132,278)(133,273)(134,276)(135,279)(136,280)
(137,283)(138,286)(139,281)(140,284)(141,287)(142,282)(143,285)(144,288);;
s2 := (  1,128)(  2,127)(  3,129)(  4,134)(  5,133)(  6,135)(  7,131)(  8,130)
(  9,132)( 10,137)( 11,136)( 12,138)( 13,143)( 14,142)( 15,144)( 16,140)
( 17,139)( 18,141)( 19,119)( 20,118)( 21,120)( 22,125)( 23,124)( 24,126)
( 25,122)( 26,121)( 27,123)( 28,110)( 29,109)( 30,111)( 31,116)( 32,115)
( 33,117)( 34,113)( 35,112)( 36,114)( 37, 74)( 38, 73)( 39, 75)( 40, 80)
( 41, 79)( 42, 81)( 43, 77)( 44, 76)( 45, 78)( 46, 83)( 47, 82)( 48, 84)
( 49, 89)( 50, 88)( 51, 90)( 52, 86)( 53, 85)( 54, 87)( 55, 92)( 56, 91)
( 57, 93)( 58, 98)( 59, 97)( 60, 99)( 61, 95)( 62, 94)( 63, 96)( 64,101)
( 65,100)( 66,102)( 67,107)( 68,106)( 69,108)( 70,104)( 71,103)( 72,105)
(145,272)(146,271)(147,273)(148,278)(149,277)(150,279)(151,275)(152,274)
(153,276)(154,281)(155,280)(156,282)(157,287)(158,286)(159,288)(160,284)
(161,283)(162,285)(163,263)(164,262)(165,264)(166,269)(167,268)(168,270)
(169,266)(170,265)(171,267)(172,254)(173,253)(174,255)(175,260)(176,259)
(177,261)(178,257)(179,256)(180,258)(181,218)(182,217)(183,219)(184,224)
(185,223)(186,225)(187,221)(188,220)(189,222)(190,227)(191,226)(192,228)
(193,233)(194,232)(195,234)(196,230)(197,229)(198,231)(199,236)(200,235)
(201,237)(202,242)(203,241)(204,243)(205,239)(206,238)(207,240)(208,245)
(209,244)(210,246)(211,251)(212,250)(213,252)(214,248)(215,247)(216,249);;
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*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;

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

```
References : None.
to this polytope