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# Polytope of Type {28,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,12}*1344c
if this polytope has a name.
Group : SmallGroup(1344,11370)
Rank : 3
Schlafli Type : {28,12}
Number of vertices, edges, etc : 56, 336, 24
Order of s0s1s2 : 42
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {28,6}*672
4-fold quotients : {28,6}*336b
7-fold quotients : {4,12}*192c
8-fold quotients : {14,6}*168
14-fold quotients : {4,6}*96
24-fold quotients : {14,2}*56
28-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {7,2}*28
56-fold quotients : {4,3}*24, {2,6}*24
112-fold quotients : {2,3}*12
168-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
```s0 := (  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)( 10, 24)
( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29, 31)( 30, 32)
( 33, 55)( 34, 56)( 35, 53)( 36, 54)( 37, 51)( 38, 52)( 39, 49)( 40, 50)
( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 57, 59)( 58, 60)( 61, 83)( 62, 84)
( 63, 81)( 64, 82)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)( 70, 76)
( 71, 73)( 72, 74)( 85, 87)( 86, 88)( 89,111)( 90,112)( 91,109)( 92,110)
( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)( 99,101)(100,102)
(113,115)(114,116)(117,139)(118,140)(119,137)(120,138)(121,135)(122,136)
(123,133)(124,134)(125,131)(126,132)(127,129)(128,130)(141,143)(142,144)
(145,167)(146,168)(147,165)(148,166)(149,163)(150,164)(151,161)(152,162)
(153,159)(154,160)(155,157)(156,158)(169,255)(170,256)(171,253)(172,254)
(173,279)(174,280)(175,277)(176,278)(177,275)(178,276)(179,273)(180,274)
(181,271)(182,272)(183,269)(184,270)(185,267)(186,268)(187,265)(188,266)
(189,263)(190,264)(191,261)(192,262)(193,259)(194,260)(195,257)(196,258)
(197,283)(198,284)(199,281)(200,282)(201,307)(202,308)(203,305)(204,306)
(205,303)(206,304)(207,301)(208,302)(209,299)(210,300)(211,297)(212,298)
(213,295)(214,296)(215,293)(216,294)(217,291)(218,292)(219,289)(220,290)
(221,287)(222,288)(223,285)(224,286)(225,311)(226,312)(227,309)(228,310)
(229,335)(230,336)(231,333)(232,334)(233,331)(234,332)(235,329)(236,330)
(237,327)(238,328)(239,325)(240,326)(241,323)(242,324)(243,321)(244,322)
(245,319)(246,320)(247,317)(248,318)(249,315)(250,316)(251,313)(252,314);;
s1 := (  1,173)(  2,174)(  3,176)(  4,175)(  5,169)(  6,170)(  7,172)(  8,171)
(  9,193)( 10,194)( 11,196)( 12,195)( 13,189)( 14,190)( 15,192)( 16,191)
( 17,185)( 18,186)( 19,188)( 20,187)( 21,181)( 22,182)( 23,184)( 24,183)
( 25,177)( 26,178)( 27,180)( 28,179)( 29,229)( 30,230)( 31,232)( 32,231)
( 33,225)( 34,226)( 35,228)( 36,227)( 37,249)( 38,250)( 39,252)( 40,251)
( 41,245)( 42,246)( 43,248)( 44,247)( 45,241)( 46,242)( 47,244)( 48,243)
( 49,237)( 50,238)( 51,240)( 52,239)( 53,233)( 54,234)( 55,236)( 56,235)
( 57,201)( 58,202)( 59,204)( 60,203)( 61,197)( 62,198)( 63,200)( 64,199)
( 65,221)( 66,222)( 67,224)( 68,223)( 69,217)( 70,218)( 71,220)( 72,219)
( 73,213)( 74,214)( 75,216)( 76,215)( 77,209)( 78,210)( 79,212)( 80,211)
( 81,205)( 82,206)( 83,208)( 84,207)( 85,257)( 86,258)( 87,260)( 88,259)
( 89,253)( 90,254)( 91,256)( 92,255)( 93,277)( 94,278)( 95,280)( 96,279)
( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)(104,271)
(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)(112,263)
(113,313)(114,314)(115,316)(116,315)(117,309)(118,310)(119,312)(120,311)
(121,333)(122,334)(123,336)(124,335)(125,329)(126,330)(127,332)(128,331)
(129,325)(130,326)(131,328)(132,327)(133,321)(134,322)(135,324)(136,323)
(137,317)(138,318)(139,320)(140,319)(141,285)(142,286)(143,288)(144,287)
(145,281)(146,282)(147,284)(148,283)(149,305)(150,306)(151,308)(152,307)
(153,301)(154,302)(155,304)(156,303)(157,297)(158,298)(159,300)(160,299)
(161,293)(162,294)(163,296)(164,295)(165,289)(166,290)(167,292)(168,291);;
s2 := (  1, 29)(  2, 32)(  3, 31)(  4, 30)(  5, 33)(  6, 36)(  7, 35)(  8, 34)
(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)( 16, 42)
( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)( 24, 50)
( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)( 70, 72)
( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)( 89,117)
( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)( 97,125)
( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)(105,133)
(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)(142,144)
(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,281)(170,284)
(171,283)(172,282)(173,285)(174,288)(175,287)(176,286)(177,289)(178,292)
(179,291)(180,290)(181,293)(182,296)(183,295)(184,294)(185,297)(186,300)
(187,299)(188,298)(189,301)(190,304)(191,303)(192,302)(193,305)(194,308)
(195,307)(196,306)(197,253)(198,256)(199,255)(200,254)(201,257)(202,260)
(203,259)(204,258)(205,261)(206,264)(207,263)(208,262)(209,265)(210,268)
(211,267)(212,266)(213,269)(214,272)(215,271)(216,270)(217,273)(218,276)
(219,275)(220,274)(221,277)(222,280)(223,279)(224,278)(225,309)(226,312)
(227,311)(228,310)(229,313)(230,316)(231,315)(232,314)(233,317)(234,320)
(235,319)(236,318)(237,321)(238,324)(239,323)(240,322)(241,325)(242,328)
(243,327)(244,326)(245,329)(246,332)(247,331)(248,330)(249,333)(250,336)
(251,335)(252,334);;
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*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*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*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;

```
Permutation Representation (Magma) :
```s0 := Sym(336)!(  1,  3)(  2,  4)(  5, 27)(  6, 28)(  7, 25)(  8, 26)(  9, 23)
( 10, 24)( 11, 21)( 12, 22)( 13, 19)( 14, 20)( 15, 17)( 16, 18)( 29, 31)
( 30, 32)( 33, 55)( 34, 56)( 35, 53)( 36, 54)( 37, 51)( 38, 52)( 39, 49)
( 40, 50)( 41, 47)( 42, 48)( 43, 45)( 44, 46)( 57, 59)( 58, 60)( 61, 83)
( 62, 84)( 63, 81)( 64, 82)( 65, 79)( 66, 80)( 67, 77)( 68, 78)( 69, 75)
( 70, 76)( 71, 73)( 72, 74)( 85, 87)( 86, 88)( 89,111)( 90,112)( 91,109)
( 92,110)( 93,107)( 94,108)( 95,105)( 96,106)( 97,103)( 98,104)( 99,101)
(100,102)(113,115)(114,116)(117,139)(118,140)(119,137)(120,138)(121,135)
(122,136)(123,133)(124,134)(125,131)(126,132)(127,129)(128,130)(141,143)
(142,144)(145,167)(146,168)(147,165)(148,166)(149,163)(150,164)(151,161)
(152,162)(153,159)(154,160)(155,157)(156,158)(169,255)(170,256)(171,253)
(172,254)(173,279)(174,280)(175,277)(176,278)(177,275)(178,276)(179,273)
(180,274)(181,271)(182,272)(183,269)(184,270)(185,267)(186,268)(187,265)
(188,266)(189,263)(190,264)(191,261)(192,262)(193,259)(194,260)(195,257)
(196,258)(197,283)(198,284)(199,281)(200,282)(201,307)(202,308)(203,305)
(204,306)(205,303)(206,304)(207,301)(208,302)(209,299)(210,300)(211,297)
(212,298)(213,295)(214,296)(215,293)(216,294)(217,291)(218,292)(219,289)
(220,290)(221,287)(222,288)(223,285)(224,286)(225,311)(226,312)(227,309)
(228,310)(229,335)(230,336)(231,333)(232,334)(233,331)(234,332)(235,329)
(236,330)(237,327)(238,328)(239,325)(240,326)(241,323)(242,324)(243,321)
(244,322)(245,319)(246,320)(247,317)(248,318)(249,315)(250,316)(251,313)
(252,314);
s1 := Sym(336)!(  1,173)(  2,174)(  3,176)(  4,175)(  5,169)(  6,170)(  7,172)
(  8,171)(  9,193)( 10,194)( 11,196)( 12,195)( 13,189)( 14,190)( 15,192)
( 16,191)( 17,185)( 18,186)( 19,188)( 20,187)( 21,181)( 22,182)( 23,184)
( 24,183)( 25,177)( 26,178)( 27,180)( 28,179)( 29,229)( 30,230)( 31,232)
( 32,231)( 33,225)( 34,226)( 35,228)( 36,227)( 37,249)( 38,250)( 39,252)
( 40,251)( 41,245)( 42,246)( 43,248)( 44,247)( 45,241)( 46,242)( 47,244)
( 48,243)( 49,237)( 50,238)( 51,240)( 52,239)( 53,233)( 54,234)( 55,236)
( 56,235)( 57,201)( 58,202)( 59,204)( 60,203)( 61,197)( 62,198)( 63,200)
( 64,199)( 65,221)( 66,222)( 67,224)( 68,223)( 69,217)( 70,218)( 71,220)
( 72,219)( 73,213)( 74,214)( 75,216)( 76,215)( 77,209)( 78,210)( 79,212)
( 80,211)( 81,205)( 82,206)( 83,208)( 84,207)( 85,257)( 86,258)( 87,260)
( 88,259)( 89,253)( 90,254)( 91,256)( 92,255)( 93,277)( 94,278)( 95,280)
( 96,279)( 97,273)( 98,274)( 99,276)(100,275)(101,269)(102,270)(103,272)
(104,271)(105,265)(106,266)(107,268)(108,267)(109,261)(110,262)(111,264)
(112,263)(113,313)(114,314)(115,316)(116,315)(117,309)(118,310)(119,312)
(120,311)(121,333)(122,334)(123,336)(124,335)(125,329)(126,330)(127,332)
(128,331)(129,325)(130,326)(131,328)(132,327)(133,321)(134,322)(135,324)
(136,323)(137,317)(138,318)(139,320)(140,319)(141,285)(142,286)(143,288)
(144,287)(145,281)(146,282)(147,284)(148,283)(149,305)(150,306)(151,308)
(152,307)(153,301)(154,302)(155,304)(156,303)(157,297)(158,298)(159,300)
(160,299)(161,293)(162,294)(163,296)(164,295)(165,289)(166,290)(167,292)
(168,291);
s2 := Sym(336)!(  1, 29)(  2, 32)(  3, 31)(  4, 30)(  5, 33)(  6, 36)(  7, 35)
(  8, 34)(  9, 37)( 10, 40)( 11, 39)( 12, 38)( 13, 41)( 14, 44)( 15, 43)
( 16, 42)( 17, 45)( 18, 48)( 19, 47)( 20, 46)( 21, 49)( 22, 52)( 23, 51)
( 24, 50)( 25, 53)( 26, 56)( 27, 55)( 28, 54)( 58, 60)( 62, 64)( 66, 68)
( 70, 72)( 74, 76)( 78, 80)( 82, 84)( 85,113)( 86,116)( 87,115)( 88,114)
( 89,117)( 90,120)( 91,119)( 92,118)( 93,121)( 94,124)( 95,123)( 96,122)
( 97,125)( 98,128)( 99,127)(100,126)(101,129)(102,132)(103,131)(104,130)
(105,133)(106,136)(107,135)(108,134)(109,137)(110,140)(111,139)(112,138)
(142,144)(146,148)(150,152)(154,156)(158,160)(162,164)(166,168)(169,281)
(170,284)(171,283)(172,282)(173,285)(174,288)(175,287)(176,286)(177,289)
(178,292)(179,291)(180,290)(181,293)(182,296)(183,295)(184,294)(185,297)
(186,300)(187,299)(188,298)(189,301)(190,304)(191,303)(192,302)(193,305)
(194,308)(195,307)(196,306)(197,253)(198,256)(199,255)(200,254)(201,257)
(202,260)(203,259)(204,258)(205,261)(206,264)(207,263)(208,262)(209,265)
(210,268)(211,267)(212,266)(213,269)(214,272)(215,271)(216,270)(217,273)
(218,276)(219,275)(220,274)(221,277)(222,280)(223,279)(224,278)(225,309)
(226,312)(227,311)(228,310)(229,313)(230,316)(231,315)(232,314)(233,317)
(234,320)(235,319)(236,318)(237,321)(238,324)(239,323)(240,322)(241,325)
(242,328)(243,327)(244,326)(245,329)(246,332)(247,331)(248,330)(249,333)
(250,336)(251,335)(252,334);
poly := sub<Sym(336)|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*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*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*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;

```
References : None.
to this polytope