Overview
- Group
- SmallGroup(768,1088556)
- Rank
- 3
- Schläfli Type
- {4,6}
- Vertices, edges, …
- 64, 192, 96
- Order of s0s1s2
- 12
- Order of s0s1s2s1
- 8
- Also known as
- if this polytope has a name.
Special Properties
- Compact Hyperbolic Quotient
- Locally Spherical
- Orientable
Quotients maximal quotients in bold
2-fold
4-fold
8-fold
16-fold
32-fold
64-fold
96-fold
Covers minimal covers in bold
None in this atlas.
Irregular Quotients of which this is a minimal cover
Click an entry to reveal its facets and vertex figures.
P/N, where N=<s0*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2*s1> of order 2
52 facets
32 vertex figures
- 32 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 2
48 facets
- 48 of {4}*8
32 vertex figures
- 32 of {6}*12
P/N, where N=<(s0*s1*s2*s1)^3*s0*s2*s1*s2> of order 2
48 facets
- 48 of {4}*8
32 vertex figures
- 32 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*(s2*s1)^2*s0*(s1*s2)^3> of order 2
48 facets
- 48 of {4}*8
32 vertex figures
- 32 of {6}*12
P/N, where N=<(s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1*s0*s2, s0*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*s2*s1> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<s0*(s1*s2)^2*s1*s0*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1> of order 4
24 facets
- 24 of {4}*8
24 vertex figures
P/N, where N=<(s1*s2)^3, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 4
24 facets
- 24 of {4}*8
20 vertex figures
P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s2*s1*s0, (s0*s1)^2*(s2*s1*s0*s1)^2> of order 4
30 facets
16 vertex figures
- 16 of {6}*12
P/N, where N=<(s0*s2*s1)^2*s0*(s1*s2)^2*s1, s0*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2*s1> of order 4
26 facets
16 vertex figures
- 16 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*s1*s2*s1*s0*s2, s0*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2*s1> of order 4
26 facets
16 vertex figures
- 16 of {6}*12
P/N, where N=<(s0*s1)^2, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 4
28 facets
16 vertex figures
- 16 of {6}*12
P/N, where N=<s0*(s1*s0*s2)^2*s1*s0*s1*s2*s1*s0*s2, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<(s0*s2*s1)^2*s0*(s1*s2)^2*s1, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2, s0*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*(s1*s2)^2> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<s0*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 4
24 facets
- 24 of {4}*8
16 vertex figures
- 16 of {6}*12
P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*(s2*s1)^2*s0*s1*s2> of order 6
16 facets
- 16 of {4}*8
12 vertex figures
P/N, where N=<(s0*s1)^2, (s0*s2*s1)^2*s0*(s1*s2)^2*s1, s0*s1*s2*s1*s0*s1*(s2*s1*s0)^2*s1*s2> of order 8
14 facets
8 vertex figures
- 8 of {6}*12
P/N, where N=<s0*(s1*s2)^2*s1*s0*s2, s0*s1*s0*s2*s1*s0*(s2*s1)^2> of order 8
12 facets
- 12 of {4}*8
12 vertex figures
P/N, where N=<(s0*s1)^2*s2*s1*s0*s2*s1*s2, (s0*s2*s1)^2*s0*(s1*s2)^2*s1> of order 8
12 facets
- 12 of {4}*8
8 vertex figures
- 8 of {6}*12
P/N, where N=<(s0*s1)^2, s0*(s1*s2)^2*s1*s0*s2> of order 8
14 facets
12 vertex figures
P/N, where N=<s0*(s1*s2)^2*s1*s0*s2, s1*s0*(s1*s2)^2*s1*s0*s2*s1, (s0*s1)^2*(s2*s1)^2*s0*s2*s1*s0> of order 8
12 facets
- 12 of {4}*8
14 vertex figures
P/N, where N=<(s1*s2)^3, (s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1*s0*s2, s0*s1*s0*(s2*s1)^2*s0*s1*s2*s1*s0*s2*s1> of order 8
12 facets
- 12 of {4}*8
10 vertex figures
P/N, where N=<(s1*s2)^2, (s0*s1)^2*(s2*s1)^2*s0*s1*s2*s1*s0*s2> of order 12
8 facets
- 8 of {4}*8
8 vertex figures
P/N, where N=<(s1*s2)^2, s0*s1*s0*s2*s1*s0*s1*s2*s1*s0> of order 12
10 facets
8 vertex figures
Representations
Permutation Representation (GAP)
s0 := ( 1,221)( 2,222)( 3,224)( 4,223)( 5,218)( 6,217)( 7,219)( 8,220)( 9,213)( 10,214)( 11,216)( 12,215)( 13,210)( 14,209)( 15,211)( 16,212)( 17,205)( 18,206)( 19,208)( 20,207)( 21,202)( 22,201)( 23,203)( 24,204)( 25,197)( 26,198)( 27,200)( 28,199)( 29,194)( 30,193)( 31,195)( 32,196)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,245)( 42,246)( 43,248)( 44,247)( 45,242)( 46,241)( 47,243)( 48,244)( 49,237)( 50,238)( 51,240)( 52,239)( 53,234)( 54,233)( 55,235)( 56,236)( 57,229)( 58,230)( 59,232)( 60,231)( 61,226)( 62,225)( 63,227)( 64,228)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,277)( 74,278)( 75,280)( 76,279)( 77,274)( 78,273)( 79,275)( 80,276)( 81,269)( 82,270)( 83,272)( 84,271)( 85,266)( 86,265)( 87,267)( 88,268)( 89,261)( 90,262)( 91,264)( 92,263)( 93,258)( 94,257)( 95,259)( 96,260)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)(105,309)(106,310)(107,312)(108,311)(109,306)(110,305)(111,307)(112,308)(113,301)(114,302)(115,304)(116,303)(117,298)(118,297)(119,299)(120,300)(121,293)(122,294)(123,296)(124,295)(125,290)(126,289)(127,291)(128,292)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)(137,341)(138,342)(139,344)(140,343)(141,338)(142,337)(143,339)(144,340)(145,333)(146,334)(147,336)(148,335)(149,330)(150,329)(151,331)(152,332)(153,325)(154,326)(155,328)(156,327)(157,322)(158,321)(159,323)(160,324)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)(169,373)(170,374)(171,376)(172,375)(173,370)(174,369)(175,371)(176,372)(177,365)(178,366)(179,368)(180,367)(181,362)(182,361)(183,363)(184,364)(185,357)(186,358)(187,360)(188,359)(189,354)(190,353)(191,355)(192,356);; s1 := ( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)(193,194)(195,196)(197,200)(198,199)(201,202)(203,204)(205,208)(206,207)(209,218)(210,217)(211,220)(212,219)(213,224)(214,223)(215,222)(216,221)(225,258)(226,257)(227,260)(228,259)(229,264)(230,263)(231,262)(232,261)(233,266)(234,265)(235,268)(236,267)(237,272)(238,271)(239,270)(240,269)(241,282)(242,281)(243,284)(244,283)(245,288)(246,287)(247,286)(248,285)(249,274)(250,273)(251,276)(252,275)(253,280)(254,279)(255,278)(256,277)(289,290)(291,292)(293,296)(294,295)(297,298)(299,300)(301,304)(302,303)(305,314)(306,313)(307,316)(308,315)(309,320)(310,319)(311,318)(312,317)(321,354)(322,353)(323,356)(324,355)(325,360)(326,359)(327,358)(328,357)(329,362)(330,361)(331,364)(332,363)(333,368)(334,367)(335,366)(336,365)(337,378)(338,377)(339,380)(340,379)(341,384)(342,383)(343,382)(344,381)(345,370)(346,369)(347,372)(348,371)(349,376)(350,375)(351,374)(352,373);; s2 := ( 1,161)( 2,162)( 3,165)( 4,166)( 5,163)( 6,164)( 7,167)( 8,168)( 9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)(193,354)(194,353)(195,358)(196,357)(197,356)(198,355)(199,360)(200,359)(201,373)(202,374)(203,369)(204,370)(205,375)(206,376)(207,371)(208,372)(209,363)(210,364)(211,367)(212,368)(213,361)(214,362)(215,365)(216,366)(217,383)(218,384)(219,379)(220,380)(221,381)(222,382)(223,377)(224,378)(225,322)(226,321)(227,326)(228,325)(229,324)(230,323)(231,328)(232,327)(233,341)(234,342)(235,337)(236,338)(237,343)(238,344)(239,339)(240,340)(241,331)(242,332)(243,335)(244,336)(245,329)(246,330)(247,333)(248,334)(249,351)(250,352)(251,347)(252,348)(253,349)(254,350)(255,345)(256,346)(257,290)(258,289)(259,294)(260,293)(261,292)(262,291)(263,296)(264,295)(265,309)(266,310)(267,305)(268,306)(269,311)(270,312)(271,307)(272,308)(273,299)(274,300)(275,303)(276,304)(277,297)(278,298)(279,301)(280,302)(281,319)(282,320)(283,315)(284,316)(285,317)(286,318)(287,313)(288,314);; 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*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(384)!( 1,221)( 2,222)( 3,224)( 4,223)( 5,218)( 6,217)( 7,219)( 8,220)( 9,213)( 10,214)( 11,216)( 12,215)( 13,210)( 14,209)( 15,211)( 16,212)( 17,205)( 18,206)( 19,208)( 20,207)( 21,202)( 22,201)( 23,203)( 24,204)( 25,197)( 26,198)( 27,200)( 28,199)( 29,194)( 30,193)( 31,195)( 32,196)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)( 41,245)( 42,246)( 43,248)( 44,247)( 45,242)( 46,241)( 47,243)( 48,244)( 49,237)( 50,238)( 51,240)( 52,239)( 53,234)( 54,233)( 55,235)( 56,236)( 57,229)( 58,230)( 59,232)( 60,231)( 61,226)( 62,225)( 63,227)( 64,228)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)( 73,277)( 74,278)( 75,280)( 76,279)( 77,274)( 78,273)( 79,275)( 80,276)( 81,269)( 82,270)( 83,272)( 84,271)( 85,266)( 86,265)( 87,267)( 88,268)( 89,261)( 90,262)( 91,264)( 92,263)( 93,258)( 94,257)( 95,259)( 96,260)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)(105,309)(106,310)(107,312)(108,311)(109,306)(110,305)(111,307)(112,308)(113,301)(114,302)(115,304)(116,303)(117,298)(118,297)(119,299)(120,300)(121,293)(122,294)(123,296)(124,295)(125,290)(126,289)(127,291)(128,292)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)(137,341)(138,342)(139,344)(140,343)(141,338)(142,337)(143,339)(144,340)(145,333)(146,334)(147,336)(148,335)(149,330)(150,329)(151,331)(152,332)(153,325)(154,326)(155,328)(156,327)(157,322)(158,321)(159,323)(160,324)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)(169,373)(170,374)(171,376)(172,375)(173,370)(174,369)(175,371)(176,372)(177,365)(178,366)(179,368)(180,367)(181,362)(182,361)(183,363)(184,364)(185,357)(186,358)(187,360)(188,359)(189,354)(190,353)(191,355)(192,356); s1 := Sym(384)!( 5, 7)( 6, 8)( 13, 15)( 14, 16)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 31)( 22, 32)( 23, 29)( 24, 30)( 33, 65)( 34, 66)( 35, 67)( 36, 68)( 37, 71)( 38, 72)( 39, 69)( 40, 70)( 41, 73)( 42, 74)( 43, 75)( 44, 76)( 45, 79)( 46, 80)( 47, 77)( 48, 78)( 49, 89)( 50, 90)( 51, 91)( 52, 92)( 53, 95)( 54, 96)( 55, 93)( 56, 94)( 57, 81)( 58, 82)( 59, 83)( 60, 84)( 61, 87)( 62, 88)( 63, 85)( 64, 86)(101,103)(102,104)(109,111)(110,112)(113,121)(114,122)(115,123)(116,124)(117,127)(118,128)(119,125)(120,126)(129,161)(130,162)(131,163)(132,164)(133,167)(134,168)(135,165)(136,166)(137,169)(138,170)(139,171)(140,172)(141,175)(142,176)(143,173)(144,174)(145,185)(146,186)(147,187)(148,188)(149,191)(150,192)(151,189)(152,190)(153,177)(154,178)(155,179)(156,180)(157,183)(158,184)(159,181)(160,182)(193,194)(195,196)(197,200)(198,199)(201,202)(203,204)(205,208)(206,207)(209,218)(210,217)(211,220)(212,219)(213,224)(214,223)(215,222)(216,221)(225,258)(226,257)(227,260)(228,259)(229,264)(230,263)(231,262)(232,261)(233,266)(234,265)(235,268)(236,267)(237,272)(238,271)(239,270)(240,269)(241,282)(242,281)(243,284)(244,283)(245,288)(246,287)(247,286)(248,285)(249,274)(250,273)(251,276)(252,275)(253,280)(254,279)(255,278)(256,277)(289,290)(291,292)(293,296)(294,295)(297,298)(299,300)(301,304)(302,303)(305,314)(306,313)(307,316)(308,315)(309,320)(310,319)(311,318)(312,317)(321,354)(322,353)(323,356)(324,355)(325,360)(326,359)(327,358)(328,357)(329,362)(330,361)(331,364)(332,363)(333,368)(334,367)(335,366)(336,365)(337,378)(338,377)(339,380)(340,379)(341,384)(342,383)(343,382)(344,381)(345,370)(346,369)(347,372)(348,371)(349,376)(350,375)(351,374)(352,373); s2 := Sym(384)!( 1,161)( 2,162)( 3,165)( 4,166)( 5,163)( 6,164)( 7,167)( 8,168)( 9,182)( 10,181)( 11,178)( 12,177)( 13,184)( 14,183)( 15,180)( 16,179)( 17,172)( 18,171)( 19,176)( 20,175)( 21,170)( 22,169)( 23,174)( 24,173)( 25,192)( 26,191)( 27,188)( 28,187)( 29,190)( 30,189)( 31,186)( 32,185)( 33,129)( 34,130)( 35,133)( 36,134)( 37,131)( 38,132)( 39,135)( 40,136)( 41,150)( 42,149)( 43,146)( 44,145)( 45,152)( 46,151)( 47,148)( 48,147)( 49,140)( 50,139)( 51,144)( 52,143)( 53,138)( 54,137)( 55,142)( 56,141)( 57,160)( 58,159)( 59,156)( 60,155)( 61,158)( 62,157)( 63,154)( 64,153)( 65, 97)( 66, 98)( 67,101)( 68,102)( 69, 99)( 70,100)( 71,103)( 72,104)( 73,118)( 74,117)( 75,114)( 76,113)( 77,120)( 78,119)( 79,116)( 80,115)( 81,108)( 82,107)( 83,112)( 84,111)( 85,106)( 86,105)( 87,110)( 88,109)( 89,128)( 90,127)( 91,124)( 92,123)( 93,126)( 94,125)( 95,122)( 96,121)(193,354)(194,353)(195,358)(196,357)(197,356)(198,355)(199,360)(200,359)(201,373)(202,374)(203,369)(204,370)(205,375)(206,376)(207,371)(208,372)(209,363)(210,364)(211,367)(212,368)(213,361)(214,362)(215,365)(216,366)(217,383)(218,384)(219,379)(220,380)(221,381)(222,382)(223,377)(224,378)(225,322)(226,321)(227,326)(228,325)(229,324)(230,323)(231,328)(232,327)(233,341)(234,342)(235,337)(236,338)(237,343)(238,344)(239,339)(240,340)(241,331)(242,332)(243,335)(244,336)(245,329)(246,330)(247,333)(248,334)(249,351)(250,352)(251,347)(252,348)(253,349)(254,350)(255,345)(256,346)(257,290)(258,289)(259,294)(260,293)(261,292)(262,291)(263,296)(264,295)(265,309)(266,310)(267,305)(268,306)(269,311)(270,312)(271,307)(272,308)(273,299)(274,300)(275,303)(276,304)(277,297)(278,298)(279,301)(280,302)(281,319)(282,320)(283,315)(284,316)(285,317)(286,318)(287,313)(288,314); poly := sub<Sym(384)|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*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s0 >;
References
None.
to this polytope.