Overview
- Group
- SmallGroup(1728,30413)
- Rank
- 4
- Schläfli Type
- {6,4,4}
- Vertices, edges, …
- 54, 108, 72, 4
- Order of s0s1s2s3
- 12
- Order of s0s1s2s3s2s1
- 2
- Also known as
- if this polytope has a name.
Special Properties
- Universal
- Orientable
- Flat
Quotients maximal quotients in bold
2-fold
3-fold
4-fold
6-fold
12-fold
27-fold
54-fold
108-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=<(s1*s2*s1*s0)^2*(s1*s2)^2> of order 2
4 facets
- 4 of 2-fold non-regular quotient of {6,4}*432a
30 vertex figures
P/N, where N=<s0*s1*(s2*s1*s0)^2*(s1*s2)^2> of order 2
4 facets
- 4 of 2-fold non-regular quotient of {6,4}*432a
27 vertex figures
- 27 of {4,4}*32
P/N, where N=<s0*s2*(s1*s0)^2*s2*s1*s0*s1> of order 3
4 facets
- 4 of 3-fold non-regular quotient of {6,4}*432a
18 vertex figures
- 18 of {4,4}*32
P/N, where N=<s1*(s2*s1*s0)^2*s2*s1*s2> of order 3
4 facets
- 4 of 3-fold non-regular quotient of {6,4}*432a
18 vertex figures
- 18 of {4,4}*32
P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1, s0*s1*(s2*s1*s0)^2*(s1*s2)^2> of order 4
4 facets
- 4 of 4-fold non-regular quotient of {6,4}*432a
15 vertex figures
P/N, where N=<(s1*s2)^2, s0*s1*s2*s1*s0*s2> of order 6
4 facets
- 4 of 6-fold non-regular quotient of {6,4}*432a
12 vertex figures
P/N, where N=<(s0*s1)^2, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6
4 facets
- 4 of 6-fold non-regular quotient of {6,4}*432a
12 vertex figures
P/N, where N=<s0*s1*s2*(s1*s0)^2*s1*s2*s1, s1*s0*s1*s2*(s1*s0)^2*s1*s2> of order 6
4 facets
- 4 of 6-fold non-regular quotient of {6,4}*432a
9 vertex figures
- 9 of {4,4}*32
Representations
Permutation Representation (GAP)
s0 := ( 1,109)( 2,110)( 3,111)( 4,117)( 5,115)( 6,116)( 7,113)( 8,114)( 9,112)( 10,129)( 11,127)( 12,128)( 13,134)( 14,135)( 15,133)( 16,130)( 17,131)( 18,132)( 19,119)( 20,120)( 21,118)( 22,124)( 23,125)( 24,126)( 25,123)( 26,121)( 27,122)( 28,136)( 29,137)( 30,138)( 31,144)( 32,142)( 33,143)( 34,140)( 35,141)( 36,139)( 37,156)( 38,154)( 39,155)( 40,161)( 41,162)( 42,160)( 43,157)( 44,158)( 45,159)( 46,146)( 47,147)( 48,145)( 49,151)( 50,152)( 51,153)( 52,150)( 53,148)( 54,149)( 55,163)( 56,164)( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)(105,207)(106,204)(107,202)(108,203);; s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)( 13, 23)( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102)(109,118)(110,120)(111,119)(112,113)(115,134)(116,133)(117,135)(121,131)(122,130)(123,132)(124,125)(128,129)(136,145)(137,147)(138,146)(139,140)(142,161)(143,160)(144,162)(148,158)(149,157)(150,159)(151,152)(155,156)(163,172)(164,174)(165,173)(166,167)(169,188)(170,187)(171,189)(175,185)(176,184)(177,186)(178,179)(182,183)(190,199)(191,201)(192,200)(193,194)(196,215)(197,214)(198,216)(202,212)(203,211)(204,213)(205,206)(209,210);; s2 := ( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)( 79, 91)( 80, 93)( 81, 92)(110,111)(112,113)(115,117)(118,133)(119,135)(120,134)(121,128)(122,127)(123,129)(124,132)(125,131)(126,130)(137,138)(139,140)(142,144)(145,160)(146,162)(147,161)(148,155)(149,154)(150,156)(151,159)(152,158)(153,157)(163,190)(164,192)(165,191)(166,194)(167,193)(168,195)(169,198)(170,197)(171,196)(172,214)(173,216)(174,215)(175,209)(176,208)(177,210)(178,213)(179,212)(180,211)(181,203)(182,202)(183,204)(184,207)(185,206)(186,205)(187,199)(188,201)(189,200);; s3 := ( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216);; poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP)
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(216)!( 1,109)( 2,110)( 3,111)( 4,117)( 5,115)( 6,116)( 7,113)( 8,114)( 9,112)( 10,129)( 11,127)( 12,128)( 13,134)( 14,135)( 15,133)( 16,130)( 17,131)( 18,132)( 19,119)( 20,120)( 21,118)( 22,124)( 23,125)( 24,126)( 25,123)( 26,121)( 27,122)( 28,136)( 29,137)( 30,138)( 31,144)( 32,142)( 33,143)( 34,140)( 35,141)( 36,139)( 37,156)( 38,154)( 39,155)( 40,161)( 41,162)( 42,160)( 43,157)( 44,158)( 45,159)( 46,146)( 47,147)( 48,145)( 49,151)( 50,152)( 51,153)( 52,150)( 53,148)( 54,149)( 55,163)( 56,164)( 57,165)( 58,171)( 59,169)( 60,170)( 61,167)( 62,168)( 63,166)( 64,183)( 65,181)( 66,182)( 67,188)( 68,189)( 69,187)( 70,184)( 71,185)( 72,186)( 73,173)( 74,174)( 75,172)( 76,178)( 77,179)( 78,180)( 79,177)( 80,175)( 81,176)( 82,190)( 83,191)( 84,192)( 85,198)( 86,196)( 87,197)( 88,194)( 89,195)( 90,193)( 91,210)( 92,208)( 93,209)( 94,215)( 95,216)( 96,214)( 97,211)( 98,212)( 99,213)(100,200)(101,201)(102,199)(103,205)(104,206)(105,207)(106,204)(107,202)(108,203); s1 := Sym(216)!( 1, 10)( 2, 12)( 3, 11)( 4, 5)( 7, 26)( 8, 25)( 9, 27)( 13, 23)( 14, 22)( 15, 24)( 16, 17)( 20, 21)( 28, 37)( 29, 39)( 30, 38)( 31, 32)( 34, 53)( 35, 52)( 36, 54)( 40, 50)( 41, 49)( 42, 51)( 43, 44)( 47, 48)( 55, 64)( 56, 66)( 57, 65)( 58, 59)( 61, 80)( 62, 79)( 63, 81)( 67, 77)( 68, 76)( 69, 78)( 70, 71)( 74, 75)( 82, 91)( 83, 93)( 84, 92)( 85, 86)( 88,107)( 89,106)( 90,108)( 94,104)( 95,103)( 96,105)( 97, 98)(101,102)(109,118)(110,120)(111,119)(112,113)(115,134)(116,133)(117,135)(121,131)(122,130)(123,132)(124,125)(128,129)(136,145)(137,147)(138,146)(139,140)(142,161)(143,160)(144,162)(148,158)(149,157)(150,159)(151,152)(155,156)(163,172)(164,174)(165,173)(166,167)(169,188)(170,187)(171,189)(175,185)(176,184)(177,186)(178,179)(182,183)(190,199)(191,201)(192,200)(193,194)(196,215)(197,214)(198,216)(202,212)(203,211)(204,213)(205,206)(209,210); s2 := Sym(216)!( 2, 3)( 4, 5)( 7, 9)( 10, 25)( 11, 27)( 12, 26)( 13, 20)( 14, 19)( 15, 21)( 16, 24)( 17, 23)( 18, 22)( 29, 30)( 31, 32)( 34, 36)( 37, 52)( 38, 54)( 39, 53)( 40, 47)( 41, 46)( 42, 48)( 43, 51)( 44, 50)( 45, 49)( 55, 82)( 56, 84)( 57, 83)( 58, 86)( 59, 85)( 60, 87)( 61, 90)( 62, 89)( 63, 88)( 64,106)( 65,108)( 66,107)( 67,101)( 68,100)( 69,102)( 70,105)( 71,104)( 72,103)( 73, 95)( 74, 94)( 75, 96)( 76, 99)( 77, 98)( 78, 97)( 79, 91)( 80, 93)( 81, 92)(110,111)(112,113)(115,117)(118,133)(119,135)(120,134)(121,128)(122,127)(123,129)(124,132)(125,131)(126,130)(137,138)(139,140)(142,144)(145,160)(146,162)(147,161)(148,155)(149,154)(150,156)(151,159)(152,158)(153,157)(163,190)(164,192)(165,191)(166,194)(167,193)(168,195)(169,198)(170,197)(171,196)(172,214)(173,216)(174,215)(175,209)(176,208)(177,210)(178,213)(179,212)(180,211)(181,203)(182,202)(183,204)(184,207)(185,206)(186,205)(187,199)(188,201)(189,200); s3 := Sym(216)!( 1, 55)( 2, 56)( 3, 57)( 4, 58)( 5, 59)( 6, 60)( 7, 61)( 8, 62)( 9, 63)( 10, 64)( 11, 65)( 12, 66)( 13, 67)( 14, 68)( 15, 69)( 16, 70)( 17, 71)( 18, 72)( 19, 73)( 20, 74)( 21, 75)( 22, 76)( 23, 77)( 24, 78)( 25, 79)( 26, 80)( 27, 81)( 28, 82)( 29, 83)( 30, 84)( 31, 85)( 32, 86)( 33, 87)( 34, 88)( 35, 89)( 36, 90)( 37, 91)( 38, 92)( 39, 93)( 40, 94)( 41, 95)( 42, 96)( 43, 97)( 44, 98)( 45, 99)( 46,100)( 47,101)( 48,102)( 49,103)( 50,104)( 51,105)( 52,106)( 53,107)( 54,108)(109,163)(110,164)(111,165)(112,166)(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)(162,216); poly := sub<Sym(216)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma)
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;
References
None.
to this polytope.