# Polytope of Type {12,3}

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

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

```
References : None.
to this polytope