Overview
- Group
- SmallGroup(1728,47874)
- Rank
- 4
- Schläfli Type
- {6,6,6}
- Vertices, edges, …
- 24, 72, 72, 6
- 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
3-fold
4-fold
6-fold
9-fold
12-fold
18-fold
24-fold
36-fold
72-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=<s0*s1*s0*s2*(s1*s0)^2*s2*s1> of order 2
6 facets
- 6 of 2-fold non-regular quotient of {6,6}*288a
12 vertex figures
- 12 of {6,6}*72a
P/N, where N=<(s0*s1)^3> of order 2
6 facets
- 6 of 2-fold non-regular quotient of {6,6}*288a
12 vertex figures
- 12 of {6,6}*72a
P/N, where N=<(s0*s1)^2> of order 3
6 facets
- 6 of 3-fold non-regular quotient of {6,6}*288a
8 vertex figures
- 8 of {6,6}*72a
P/N, where N=<(s1*s0)^2*s2*s1*s0*s1*s2> of order 4
6 facets
- 6 of 4-fold non-regular quotient of {6,6}*288a
6 vertex figures
- 6 of {6,6}*72a
Representations
Permutation Representation (GAP)
s0 := ( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215);; s1 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 61)( 14, 64)( 15, 63)( 16, 62)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)( 24, 70)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 53)( 30, 56)( 31, 55)( 32, 54)( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 74, 76)( 78, 80)( 82, 84)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,101)( 90,104)( 91,103)( 92,102)( 93,105)( 94,108)( 95,107)( 96,106)(109,145)(110,148)(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,169)(122,172)(123,171)(124,170)(125,173)(126,176)(127,175)(128,174)(129,177)(130,180)(131,179)(132,178)(133,157)(134,160)(135,159)(136,158)(137,161)(138,164)(139,163)(140,162)(141,165)(142,168)(143,167)(144,166)(182,184)(186,188)(190,192)(193,205)(194,208)(195,207)(196,206)(197,209)(198,212)(199,211)(200,210)(201,213)(202,216)(203,215)(204,214);; s2 := ( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 22)( 6, 21)( 7, 23)( 8, 24)( 9, 18)( 10, 17)( 11, 19)( 12, 20)( 25, 26)( 29, 34)( 30, 33)( 31, 35)( 32, 36)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 94)( 42, 93)( 43, 95)( 44, 96)( 45, 90)( 46, 89)( 47, 91)( 48, 92)( 49, 74)( 50, 73)( 51, 75)( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)( 58, 77)( 59, 79)( 60, 80)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,106)( 66,105)( 67,107)( 68,108)( 69,102)( 70,101)( 71,103)( 72,104)(109,122)(110,121)(111,123)(112,124)(113,130)(114,129)(115,131)(116,132)(117,126)(118,125)(119,127)(120,128)(133,134)(137,142)(138,141)(139,143)(140,144)(145,194)(146,193)(147,195)(148,196)(149,202)(150,201)(151,203)(152,204)(153,198)(154,197)(155,199)(156,200)(157,182)(158,181)(159,183)(160,184)(161,190)(162,189)(163,191)(164,192)(165,186)(166,185)(167,187)(168,188)(169,206)(170,205)(171,207)(172,208)(173,214)(174,213)(175,215)(176,216)(177,210)(178,209)(179,211)(180,212);; s3 := ( 1,113)( 2,114)( 3,115)( 4,116)( 5,109)( 6,110)( 7,111)( 8,112)( 9,117)( 10,118)( 11,119)( 12,120)( 13,125)( 14,126)( 15,127)( 16,128)( 17,121)( 18,122)( 19,123)( 20,124)( 21,129)( 22,130)( 23,131)( 24,132)( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)( 33,141)( 34,142)( 35,143)( 36,144)( 37,149)( 38,150)( 39,151)( 40,152)( 41,145)( 42,146)( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,156)( 49,161)( 50,162)( 51,163)( 52,164)( 53,157)( 54,158)( 55,159)( 56,160)( 57,165)( 58,166)( 59,167)( 60,168)( 61,173)( 62,174)( 63,175)( 64,176)( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)( 73,185)( 74,186)( 75,187)( 76,188)( 77,181)( 78,182)( 79,183)( 80,184)( 81,189)( 82,190)( 83,191)( 84,192)( 85,197)( 86,198)( 87,199)( 88,200)( 89,193)( 90,194)( 91,195)( 92,196)( 93,201)( 94,202)( 95,203)( 96,204)( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)(105,213)(106,214)(107,215)(108,216);; 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*s3*s2*s1*s2*s3*s2,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3,
s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma)
s0 := Sym(216)!( 3, 4)( 7, 8)( 11, 12)( 15, 16)( 19, 20)( 23, 24)( 27, 28)( 31, 32)( 35, 36)( 37, 73)( 38, 74)( 39, 76)( 40, 75)( 41, 77)( 42, 78)( 43, 80)( 44, 79)( 45, 81)( 46, 82)( 47, 84)( 48, 83)( 49, 85)( 50, 86)( 51, 88)( 52, 87)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 93)( 58, 94)( 59, 96)( 60, 95)( 61, 97)( 62, 98)( 63,100)( 64, 99)( 65,101)( 66,102)( 67,104)( 68,103)( 69,105)( 70,106)( 71,108)( 72,107)(111,112)(115,116)(119,120)(123,124)(127,128)(131,132)(135,136)(139,140)(143,144)(145,181)(146,182)(147,184)(148,183)(149,185)(150,186)(151,188)(152,187)(153,189)(154,190)(155,192)(156,191)(157,193)(158,194)(159,196)(160,195)(161,197)(162,198)(163,200)(164,199)(165,201)(166,202)(167,204)(168,203)(169,205)(170,206)(171,208)(172,207)(173,209)(174,210)(175,212)(176,211)(177,213)(178,214)(179,216)(180,215); s1 := Sym(216)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 41)( 6, 44)( 7, 43)( 8, 42)( 9, 45)( 10, 48)( 11, 47)( 12, 46)( 13, 61)( 14, 64)( 15, 63)( 16, 62)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 69)( 22, 72)( 23, 71)( 24, 70)( 25, 49)( 26, 52)( 27, 51)( 28, 50)( 29, 53)( 30, 56)( 31, 55)( 32, 54)( 33, 57)( 34, 60)( 35, 59)( 36, 58)( 74, 76)( 78, 80)( 82, 84)( 85, 97)( 86,100)( 87, 99)( 88, 98)( 89,101)( 90,104)( 91,103)( 92,102)( 93,105)( 94,108)( 95,107)( 96,106)(109,145)(110,148)(111,147)(112,146)(113,149)(114,152)(115,151)(116,150)(117,153)(118,156)(119,155)(120,154)(121,169)(122,172)(123,171)(124,170)(125,173)(126,176)(127,175)(128,174)(129,177)(130,180)(131,179)(132,178)(133,157)(134,160)(135,159)(136,158)(137,161)(138,164)(139,163)(140,162)(141,165)(142,168)(143,167)(144,166)(182,184)(186,188)(190,192)(193,205)(194,208)(195,207)(196,206)(197,209)(198,212)(199,211)(200,210)(201,213)(202,216)(203,215)(204,214); s2 := Sym(216)!( 1, 14)( 2, 13)( 3, 15)( 4, 16)( 5, 22)( 6, 21)( 7, 23)( 8, 24)( 9, 18)( 10, 17)( 11, 19)( 12, 20)( 25, 26)( 29, 34)( 30, 33)( 31, 35)( 32, 36)( 37, 86)( 38, 85)( 39, 87)( 40, 88)( 41, 94)( 42, 93)( 43, 95)( 44, 96)( 45, 90)( 46, 89)( 47, 91)( 48, 92)( 49, 74)( 50, 73)( 51, 75)( 52, 76)( 53, 82)( 54, 81)( 55, 83)( 56, 84)( 57, 78)( 58, 77)( 59, 79)( 60, 80)( 61, 98)( 62, 97)( 63, 99)( 64,100)( 65,106)( 66,105)( 67,107)( 68,108)( 69,102)( 70,101)( 71,103)( 72,104)(109,122)(110,121)(111,123)(112,124)(113,130)(114,129)(115,131)(116,132)(117,126)(118,125)(119,127)(120,128)(133,134)(137,142)(138,141)(139,143)(140,144)(145,194)(146,193)(147,195)(148,196)(149,202)(150,201)(151,203)(152,204)(153,198)(154,197)(155,199)(156,200)(157,182)(158,181)(159,183)(160,184)(161,190)(162,189)(163,191)(164,192)(165,186)(166,185)(167,187)(168,188)(169,206)(170,205)(171,207)(172,208)(173,214)(174,213)(175,215)(176,216)(177,210)(178,209)(179,211)(180,212); s3 := Sym(216)!( 1,113)( 2,114)( 3,115)( 4,116)( 5,109)( 6,110)( 7,111)( 8,112)( 9,117)( 10,118)( 11,119)( 12,120)( 13,125)( 14,126)( 15,127)( 16,128)( 17,121)( 18,122)( 19,123)( 20,124)( 21,129)( 22,130)( 23,131)( 24,132)( 25,137)( 26,138)( 27,139)( 28,140)( 29,133)( 30,134)( 31,135)( 32,136)( 33,141)( 34,142)( 35,143)( 36,144)( 37,149)( 38,150)( 39,151)( 40,152)( 41,145)( 42,146)( 43,147)( 44,148)( 45,153)( 46,154)( 47,155)( 48,156)( 49,161)( 50,162)( 51,163)( 52,164)( 53,157)( 54,158)( 55,159)( 56,160)( 57,165)( 58,166)( 59,167)( 60,168)( 61,173)( 62,174)( 63,175)( 64,176)( 65,169)( 66,170)( 67,171)( 68,172)( 69,177)( 70,178)( 71,179)( 72,180)( 73,185)( 74,186)( 75,187)( 76,188)( 77,181)( 78,182)( 79,183)( 80,184)( 81,189)( 82,190)( 83,191)( 84,192)( 85,197)( 86,198)( 87,199)( 88,200)( 89,193)( 90,194)( 91,195)( 92,196)( 93,201)( 94,202)( 95,203)( 96,204)( 97,209)( 98,210)( 99,211)(100,212)(101,205)(102,206)(103,207)(104,208)(105,213)(106,214)(107,215)(108,216); 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*s3*s2*s1*s2*s3*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1 >;
References
None.
to this polytope.