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