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# Polytope of Type {3,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,12}*288
if this polytope has a name.
Group : SmallGroup(288,847)
Rank : 3
Schlafli Type : {3,12}
Number of vertices, edges, etc : 12, 72, 48
Order of s0s1s2 : 24
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
{3,12,2} of size 576
{3,12,4} of size 1152
{3,12,3} of size 1440
{3,12,6} of size 1728
Vertex Figure Of :
{2,3,12} of size 576
{4,3,12} of size 1152
{6,3,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {3,6}*144
3-fold quotients : {3,12}*96
6-fold quotients : {3,6}*48
8-fold quotients : {3,6}*36
12-fold quotients : {3,3}*24
24-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
2-fold covers : {6,12}*576c
3-fold covers : {9,12}*864, {3,12}*864
4-fold covers : {3,12}*1152a, {12,12}*1152e, {12,12}*1152g, {6,12}*1152e, {3,12}*1152b
5-fold covers : {15,12}*1440c
6-fold covers : {18,12}*1728a, {6,12}*1728c, {6,12}*1728g
Permutation Representation (GAP) :
```s0 := (  1,217)(  2,218)(  3,221)(  4,222)(  5,219)(  6,220)(  7,224)(  8,223)
(  9,233)( 10,234)( 11,237)( 12,238)( 13,235)( 14,236)( 15,240)( 16,239)
( 17,225)( 18,226)( 19,229)( 20,230)( 21,227)( 22,228)( 23,232)( 24,231)
( 25,265)( 26,266)( 27,269)( 28,270)( 29,267)( 30,268)( 31,272)( 32,271)
( 33,281)( 34,282)( 35,285)( 36,286)( 37,283)( 38,284)( 39,288)( 40,287)
( 41,273)( 42,274)( 43,277)( 44,278)( 45,275)( 46,276)( 47,280)( 48,279)
( 49,241)( 50,242)( 51,245)( 52,246)( 53,243)( 54,244)( 55,248)( 56,247)
( 57,257)( 58,258)( 59,261)( 60,262)( 61,259)( 62,260)( 63,264)( 64,263)
( 65,249)( 66,250)( 67,253)( 68,254)( 69,251)( 70,252)( 71,256)( 72,255)
( 73,146)( 74,145)( 75,150)( 76,149)( 77,148)( 78,147)( 79,151)( 80,152)
( 81,162)( 82,161)( 83,166)( 84,165)( 85,164)( 86,163)( 87,167)( 88,168)
( 89,154)( 90,153)( 91,158)( 92,157)( 93,156)( 94,155)( 95,159)( 96,160)
( 97,194)( 98,193)( 99,198)(100,197)(101,196)(102,195)(103,199)(104,200)
(105,210)(106,209)(107,214)(108,213)(109,212)(110,211)(111,215)(112,216)
(113,202)(114,201)(115,206)(116,205)(117,204)(118,203)(119,207)(120,208)
(121,170)(122,169)(123,174)(124,173)(125,172)(126,171)(127,175)(128,176)
(129,186)(130,185)(131,190)(132,189)(133,188)(134,187)(135,191)(136,192)
(137,178)(138,177)(139,182)(140,181)(141,180)(142,179)(143,183)(144,184);;
s1 := (  1,249)(  2,250)(  3,252)(  4,251)(  5,255)(  6,256)(  7,253)(  8,254)
(  9,241)( 10,242)( 11,244)( 12,243)( 13,247)( 14,248)( 15,245)( 16,246)
( 17,257)( 18,258)( 19,260)( 20,259)( 21,263)( 22,264)( 23,261)( 24,262)
( 25,225)( 26,226)( 27,228)( 28,227)( 29,231)( 30,232)( 31,229)( 32,230)
( 33,217)( 34,218)( 35,220)( 36,219)( 37,223)( 38,224)( 39,221)( 40,222)
( 41,233)( 42,234)( 43,236)( 44,235)( 45,239)( 46,240)( 47,237)( 48,238)
( 49,273)( 50,274)( 51,276)( 52,275)( 53,279)( 54,280)( 55,277)( 56,278)
( 57,265)( 58,266)( 59,268)( 60,267)( 61,271)( 62,272)( 63,269)( 64,270)
( 65,281)( 66,282)( 67,284)( 68,283)( 69,287)( 70,288)( 71,285)( 72,286)
( 73,178)( 74,177)( 75,179)( 76,180)( 77,184)( 78,183)( 79,182)( 80,181)
( 81,170)( 82,169)( 83,171)( 84,172)( 85,176)( 86,175)( 87,174)( 88,173)
( 89,186)( 90,185)( 91,187)( 92,188)( 93,192)( 94,191)( 95,190)( 96,189)
( 97,154)( 98,153)( 99,155)(100,156)(101,160)(102,159)(103,158)(104,157)
(105,146)(106,145)(107,147)(108,148)(109,152)(110,151)(111,150)(112,149)
(113,162)(114,161)(115,163)(116,164)(117,168)(118,167)(119,166)(120,165)
(121,202)(122,201)(123,203)(124,204)(125,208)(126,207)(127,206)(128,205)
(129,194)(130,193)(131,195)(132,196)(133,200)(134,199)(135,198)(136,197)
(137,210)(138,209)(139,211)(140,212)(141,216)(142,215)(143,214)(144,213);;
s2 := (  1, 79)(  2, 80)(  3, 76)(  4, 75)(  5, 77)(  6, 78)(  7, 73)(  8, 74)
(  9, 87)( 10, 88)( 11, 84)( 12, 83)( 13, 85)( 14, 86)( 15, 81)( 16, 82)
( 17, 95)( 18, 96)( 19, 92)( 20, 91)( 21, 93)( 22, 94)( 23, 89)( 24, 90)
( 25,127)( 26,128)( 27,124)( 28,123)( 29,125)( 30,126)( 31,121)( 32,122)
( 33,135)( 34,136)( 35,132)( 36,131)( 37,133)( 38,134)( 39,129)( 40,130)
( 41,143)( 42,144)( 43,140)( 44,139)( 45,141)( 46,142)( 47,137)( 48,138)
( 49,103)( 50,104)( 51,100)( 52, 99)( 53,101)( 54,102)( 55, 97)( 56, 98)
( 57,111)( 58,112)( 59,108)( 60,107)( 61,109)( 62,110)( 63,105)( 64,106)
( 65,119)( 66,120)( 67,116)( 68,115)( 69,117)( 70,118)( 71,113)( 72,114)
(145,223)(146,224)(147,220)(148,219)(149,221)(150,222)(151,217)(152,218)
(153,231)(154,232)(155,228)(156,227)(157,229)(158,230)(159,225)(160,226)
(161,239)(162,240)(163,236)(164,235)(165,237)(166,238)(167,233)(168,234)
(169,271)(170,272)(171,268)(172,267)(173,269)(174,270)(175,265)(176,266)
(177,279)(178,280)(179,276)(180,275)(181,277)(182,278)(183,273)(184,274)
(185,287)(186,288)(187,284)(188,283)(189,285)(190,286)(191,281)(192,282)
(193,247)(194,248)(195,244)(196,243)(197,245)(198,246)(199,241)(200,242)
(201,255)(202,256)(203,252)(204,251)(205,253)(206,254)(207,249)(208,250)
(209,263)(210,264)(211,260)(212,259)(213,261)(214,262)(215,257)(216,258);;
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, s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

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

```
References : None.
to this polytope