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# Polytope of Type {12,9}

Atlas Canonical Name : {12,9}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1789)
Rank : 3
Schlafli Type : {12,9}
Number of vertices, edges, etc : 72, 324, 54
Order of s0s1s2 : 18
Order of s0s1s2s1 : 12
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {12,9}*432, {12,3}*432
4-fold quotients : {6,9}*324a
9-fold quotients : {4,9}*144, {12,3}*144
12-fold quotients : {6,9}*108, {6,3}*108
18-fold quotients : {4,9}*72
27-fold quotients : {4,3}*48
36-fold quotients : {2,9}*36, {6,3}*36
54-fold quotients : {4,3}*24
108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 15)( 14, 16)
( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 27)( 26, 28)( 29, 35)( 30, 36)
( 31, 33)( 32, 34)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)( 42, 84)
( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 87)( 50, 88)
( 51, 85)( 52, 86)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 91)( 58, 92)
( 59, 89)( 60, 90)( 61, 99)( 62,100)( 63, 97)( 64, 98)( 65,107)( 66,108)
( 67,105)( 68,106)( 69,103)( 70,104)( 71,101)( 72,102)(109,111)(110,112)
(113,119)(114,120)(115,117)(116,118)(121,123)(122,124)(125,131)(126,132)
(127,129)(128,130)(133,135)(134,136)(137,143)(138,144)(139,141)(140,142)
(145,183)(146,184)(147,181)(148,182)(149,191)(150,192)(151,189)(152,190)
(153,187)(154,188)(155,185)(156,186)(157,195)(158,196)(159,193)(160,194)
(161,203)(162,204)(163,201)(164,202)(165,199)(166,200)(167,197)(168,198)
(169,207)(170,208)(171,205)(172,206)(173,215)(174,216)(175,213)(176,214)
(177,211)(178,212)(179,209)(180,210)(217,219)(218,220)(221,227)(222,228)
(223,225)(224,226)(229,231)(230,232)(233,239)(234,240)(235,237)(236,238)
(241,243)(242,244)(245,251)(246,252)(247,249)(248,250)(253,291)(254,292)
(255,289)(256,290)(257,299)(258,300)(259,297)(260,298)(261,295)(262,296)
(263,293)(264,294)(265,303)(266,304)(267,301)(268,302)(269,311)(270,312)
(271,309)(272,310)(273,307)(274,308)(275,305)(276,306)(277,315)(278,316)
(279,313)(280,314)(281,323)(282,324)(283,321)(284,322)(285,319)(286,320)
(287,317)(288,318);;
s1 := (  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 41)(  6, 42)(  7, 44)(  8, 43)
(  9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 61)( 14, 62)( 15, 64)( 16, 63)
( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)( 24, 71)
( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 53)( 30, 54)( 31, 56)( 32, 55)
( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 75, 76)( 79, 80)( 83, 84)( 85, 97)
( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)( 93,105)
( 94,106)( 95,108)( 96,107)(109,277)(110,278)(111,280)(112,279)(113,281)
(114,282)(115,284)(116,283)(117,285)(118,286)(119,288)(120,287)(121,265)
(122,266)(123,268)(124,267)(125,269)(126,270)(127,272)(128,271)(129,273)
(130,274)(131,276)(132,275)(133,253)(134,254)(135,256)(136,255)(137,257)
(138,258)(139,260)(140,259)(141,261)(142,262)(143,264)(144,263)(145,241)
(146,242)(147,244)(148,243)(149,245)(150,246)(151,248)(152,247)(153,249)
(154,250)(155,252)(156,251)(157,229)(158,230)(159,232)(160,231)(161,233)
(162,234)(163,236)(164,235)(165,237)(166,238)(167,240)(168,239)(169,217)
(170,218)(171,220)(172,219)(173,221)(174,222)(175,224)(176,223)(177,225)
(178,226)(179,228)(180,227)(181,313)(182,314)(183,316)(184,315)(185,317)
(186,318)(187,320)(188,319)(189,321)(190,322)(191,324)(192,323)(193,301)
(194,302)(195,304)(196,303)(197,305)(198,306)(199,308)(200,307)(201,309)
(202,310)(203,312)(204,311)(205,289)(206,290)(207,292)(208,291)(209,293)
(210,294)(211,296)(212,295)(213,297)(214,298)(215,300)(216,299);;
s2 := (  1,109)(  2,112)(  3,111)(  4,110)(  5,113)(  6,116)(  7,115)(  8,114)
(  9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)( 16,134)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)( 24,142)
( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)( 40,190)
( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)( 48,186)
( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)( 56,206)
( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)( 64,202)
( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)( 72,198)
( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)( 80,154)
( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)( 88,174)
( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)( 96,170)
( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)(104,166)
(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)(220,242)
(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)(228,250)
(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)(257,313)
(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)(265,309)
(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)(273,305)
(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)(281,289)
(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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(324)!(  1,  3)(  2,  4)(  5, 11)(  6, 12)(  7,  9)(  8, 10)( 13, 15)
( 14, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 27)( 26, 28)( 29, 35)
( 30, 36)( 31, 33)( 32, 34)( 37, 75)( 38, 76)( 39, 73)( 40, 74)( 41, 83)
( 42, 84)( 43, 81)( 44, 82)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 87)
( 50, 88)( 51, 85)( 52, 86)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 91)
( 58, 92)( 59, 89)( 60, 90)( 61, 99)( 62,100)( 63, 97)( 64, 98)( 65,107)
( 66,108)( 67,105)( 68,106)( 69,103)( 70,104)( 71,101)( 72,102)(109,111)
(110,112)(113,119)(114,120)(115,117)(116,118)(121,123)(122,124)(125,131)
(126,132)(127,129)(128,130)(133,135)(134,136)(137,143)(138,144)(139,141)
(140,142)(145,183)(146,184)(147,181)(148,182)(149,191)(150,192)(151,189)
(152,190)(153,187)(154,188)(155,185)(156,186)(157,195)(158,196)(159,193)
(160,194)(161,203)(162,204)(163,201)(164,202)(165,199)(166,200)(167,197)
(168,198)(169,207)(170,208)(171,205)(172,206)(173,215)(174,216)(175,213)
(176,214)(177,211)(178,212)(179,209)(180,210)(217,219)(218,220)(221,227)
(222,228)(223,225)(224,226)(229,231)(230,232)(233,239)(234,240)(235,237)
(236,238)(241,243)(242,244)(245,251)(246,252)(247,249)(248,250)(253,291)
(254,292)(255,289)(256,290)(257,299)(258,300)(259,297)(260,298)(261,295)
(262,296)(263,293)(264,294)(265,303)(266,304)(267,301)(268,302)(269,311)
(270,312)(271,309)(272,310)(273,307)(274,308)(275,305)(276,306)(277,315)
(278,316)(279,313)(280,314)(281,323)(282,324)(283,321)(284,322)(285,319)
(286,320)(287,317)(288,318);
s1 := Sym(324)!(  1, 37)(  2, 38)(  3, 40)(  4, 39)(  5, 41)(  6, 42)(  7, 44)
(  8, 43)(  9, 45)( 10, 46)( 11, 48)( 12, 47)( 13, 61)( 14, 62)( 15, 64)
( 16, 63)( 17, 65)( 18, 66)( 19, 68)( 20, 67)( 21, 69)( 22, 70)( 23, 72)
( 24, 71)( 25, 49)( 26, 50)( 27, 52)( 28, 51)( 29, 53)( 30, 54)( 31, 56)
( 32, 55)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 75, 76)( 79, 80)( 83, 84)
( 85, 97)( 86, 98)( 87,100)( 88, 99)( 89,101)( 90,102)( 91,104)( 92,103)
( 93,105)( 94,106)( 95,108)( 96,107)(109,277)(110,278)(111,280)(112,279)
(113,281)(114,282)(115,284)(116,283)(117,285)(118,286)(119,288)(120,287)
(121,265)(122,266)(123,268)(124,267)(125,269)(126,270)(127,272)(128,271)
(129,273)(130,274)(131,276)(132,275)(133,253)(134,254)(135,256)(136,255)
(137,257)(138,258)(139,260)(140,259)(141,261)(142,262)(143,264)(144,263)
(145,241)(146,242)(147,244)(148,243)(149,245)(150,246)(151,248)(152,247)
(153,249)(154,250)(155,252)(156,251)(157,229)(158,230)(159,232)(160,231)
(161,233)(162,234)(163,236)(164,235)(165,237)(166,238)(167,240)(168,239)
(169,217)(170,218)(171,220)(172,219)(173,221)(174,222)(175,224)(176,223)
(177,225)(178,226)(179,228)(180,227)(181,313)(182,314)(183,316)(184,315)
(185,317)(186,318)(187,320)(188,319)(189,321)(190,322)(191,324)(192,323)
(193,301)(194,302)(195,304)(196,303)(197,305)(198,306)(199,308)(200,307)
(201,309)(202,310)(203,312)(204,311)(205,289)(206,290)(207,292)(208,291)
(209,293)(210,294)(211,296)(212,295)(213,297)(214,298)(215,300)(216,299);
s2 := Sym(324)!(  1,109)(  2,112)(  3,111)(  4,110)(  5,113)(  6,116)(  7,115)
(  8,114)(  9,117)( 10,120)( 11,119)( 12,118)( 13,133)( 14,136)( 15,135)
( 16,134)( 17,137)( 18,140)( 19,139)( 20,138)( 21,141)( 22,144)( 23,143)
( 24,142)( 25,121)( 26,124)( 27,123)( 28,122)( 29,125)( 30,128)( 31,127)
( 32,126)( 33,129)( 34,132)( 35,131)( 36,130)( 37,189)( 38,192)( 39,191)
( 40,190)( 41,181)( 42,184)( 43,183)( 44,182)( 45,185)( 46,188)( 47,187)
( 48,186)( 49,213)( 50,216)( 51,215)( 52,214)( 53,205)( 54,208)( 55,207)
( 56,206)( 57,209)( 58,212)( 59,211)( 60,210)( 61,201)( 62,204)( 63,203)
( 64,202)( 65,193)( 66,196)( 67,195)( 68,194)( 69,197)( 70,200)( 71,199)
( 72,198)( 73,149)( 74,152)( 75,151)( 76,150)( 77,153)( 78,156)( 79,155)
( 80,154)( 81,145)( 82,148)( 83,147)( 84,146)( 85,173)( 86,176)( 87,175)
( 88,174)( 89,177)( 90,180)( 91,179)( 92,178)( 93,169)( 94,172)( 95,171)
( 96,170)( 97,161)( 98,164)( 99,163)(100,162)(101,165)(102,168)(103,167)
(104,166)(105,157)(106,160)(107,159)(108,158)(217,241)(218,244)(219,243)
(220,242)(221,245)(222,248)(223,247)(224,246)(225,249)(226,252)(227,251)
(228,250)(230,232)(234,236)(238,240)(253,321)(254,324)(255,323)(256,322)
(257,313)(258,316)(259,315)(260,314)(261,317)(262,320)(263,319)(264,318)
(265,309)(266,312)(267,311)(268,310)(269,301)(270,304)(271,303)(272,302)
(273,305)(274,308)(275,307)(276,306)(277,297)(278,300)(279,299)(280,298)
(281,289)(282,292)(283,291)(284,290)(285,293)(286,296)(287,295)(288,294);
poly := sub<Sym(324)|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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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