Polytope of Type {4,12,6} with group SmallGroup(1728,30413)

Overview

Group: SmallGroup(1728,30413)
Rank: 4
Schläfli Type: {4,12,6}
Vertices, edges, …: 4, 72, 108, 18
Order of s₀s₁s₂s₃: 4
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

{4,4,6}*576

4-fold

{2,12,6}*432a

6-fold

12-fold

{2,4,6}*144

27-fold

{4,4,2}*64

54-fold

108-fold

{2,2,2}*16

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*s1*s3*s2*s1*s2*s3> of order 3

6 facets

6 of {4,12}*96a

4 vertex figures

4 of 3-fold non-regular quotient of {12,6}*432f

Representations

Permutation Representation (GAP)

s0 := (  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);;
s1 := (  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);;
s2 := (  1,  2)(  4, 23)(  5, 22)(  6, 24)(  7, 16)(  8, 18)(  9, 17)( 10, 20)( 11, 19)( 12, 21)( 13, 14)( 26, 27)( 28, 29)( 31, 50)( 32, 49)( 33, 51)( 34, 43)( 35, 45)( 36, 44)( 37, 47)( 38, 46)( 39, 48)( 40, 41)( 53, 54)( 55, 56)( 58, 77)( 59, 76)( 60, 78)( 61, 70)( 62, 72)( 63, 71)( 64, 74)( 65, 73)( 66, 75)( 67, 68)( 80, 81)( 82, 83)( 85,104)( 86,103)( 87,105)( 88, 97)( 89, 99)( 90, 98)( 91,101)( 92,100)( 93,102)( 94, 95)(107,108)(109,110)(112,131)(113,130)(114,132)(115,124)(116,126)(117,125)(118,128)(119,127)(120,129)(121,122)(134,135)(136,137)(139,158)(140,157)(141,159)(142,151)(143,153)(144,152)(145,155)(146,154)(147,156)(148,149)(161,162)(163,164)(166,185)(167,184)(168,186)(169,178)(170,180)(171,179)(172,182)(173,181)(174,183)(175,176)(188,189)(190,191)(193,212)(194,211)(195,213)(196,205)(197,207)(198,206)(199,209)(200,208)(201,210)(202,203)(215,216);;
s3 := (  1,116)(  2,117)(  3,115)(  4,112)(  5,113)(  6,114)(  7,111)(  8,109)(  9,110)( 10,133)( 11,134)( 12,135)( 13,132)( 14,130)( 15,131)( 16,128)( 17,129)( 18,127)( 19,126)( 20,124)( 21,125)( 22,122)( 23,123)( 24,121)( 25,118)( 26,119)( 27,120)( 28,143)( 29,144)( 30,142)( 31,139)( 32,140)( 33,141)( 34,138)( 35,136)( 36,137)( 37,160)( 38,161)( 39,162)( 40,159)( 41,157)( 42,158)( 43,155)( 44,156)( 45,154)( 46,153)( 47,151)( 48,152)( 49,149)( 50,150)( 51,148)( 52,145)( 53,146)( 54,147)( 55,170)( 56,171)( 57,169)( 58,166)( 59,167)( 60,168)( 61,165)( 62,163)( 63,164)( 64,187)( 65,188)( 66,189)( 67,186)( 68,184)( 69,185)( 70,182)( 71,183)( 72,181)( 73,180)( 74,178)( 75,179)( 76,176)( 77,177)( 78,175)( 79,172)( 80,173)( 81,174)( 82,197)( 83,198)( 84,196)( 85,193)( 86,194)( 87,195)( 88,192)( 89,190)( 90,191)( 91,214)( 92,215)( 93,216)( 94,213)( 95,211)( 96,212)( 97,209)( 98,210)( 99,208)(100,207)(101,205)(102,206)(103,203)(104,204)(105,202)(106,199)(107,200)(108,201);;
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*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := 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);
s1 := 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);
s2 := Sym(216)!(  1,  2)(  4, 23)(  5, 22)(  6, 24)(  7, 16)(  8, 18)(  9, 17)( 10, 20)( 11, 19)( 12, 21)( 13, 14)( 26, 27)( 28, 29)( 31, 50)( 32, 49)( 33, 51)( 34, 43)( 35, 45)( 36, 44)( 37, 47)( 38, 46)( 39, 48)( 40, 41)( 53, 54)( 55, 56)( 58, 77)( 59, 76)( 60, 78)( 61, 70)( 62, 72)( 63, 71)( 64, 74)( 65, 73)( 66, 75)( 67, 68)( 80, 81)( 82, 83)( 85,104)( 86,103)( 87,105)( 88, 97)( 89, 99)( 90, 98)( 91,101)( 92,100)( 93,102)( 94, 95)(107,108)(109,110)(112,131)(113,130)(114,132)(115,124)(116,126)(117,125)(118,128)(119,127)(120,129)(121,122)(134,135)(136,137)(139,158)(140,157)(141,159)(142,151)(143,153)(144,152)(145,155)(146,154)(147,156)(148,149)(161,162)(163,164)(166,185)(167,184)(168,186)(169,178)(170,180)(171,179)(172,182)(173,181)(174,183)(175,176)(188,189)(190,191)(193,212)(194,211)(195,213)(196,205)(197,207)(198,206)(199,209)(200,208)(201,210)(202,203)(215,216);
s3 := Sym(216)!(  1,116)(  2,117)(  3,115)(  4,112)(  5,113)(  6,114)(  7,111)(  8,109)(  9,110)( 10,133)( 11,134)( 12,135)( 13,132)( 14,130)( 15,131)( 16,128)( 17,129)( 18,127)( 19,126)( 20,124)( 21,125)( 22,122)( 23,123)( 24,121)( 25,118)( 26,119)( 27,120)( 28,143)( 29,144)( 30,142)( 31,139)( 32,140)( 33,141)( 34,138)( 35,136)( 36,137)( 37,160)( 38,161)( 39,162)( 40,159)( 41,157)( 42,158)( 43,155)( 44,156)( 45,154)( 46,153)( 47,151)( 48,152)( 49,149)( 50,150)( 51,148)( 52,145)( 53,146)( 54,147)( 55,170)( 56,171)( 57,169)( 58,166)( 59,167)( 60,168)( 61,165)( 62,163)( 63,164)( 64,187)( 65,188)( 66,189)( 67,186)( 68,184)( 69,185)( 70,182)( 71,183)( 72,181)( 73,180)( 74,178)( 75,179)( 76,176)( 77,177)( 78,175)( 79,172)( 80,173)( 81,174)( 82,197)( 83,198)( 84,196)( 85,193)( 86,194)( 87,195)( 88,192)( 89,190)( 90,191)( 91,214)( 92,215)( 93,216)( 94,213)( 95,211)( 96,212)( 97,209)( 98,210)( 99,208)(100,207)(101,205)(102,206)(103,203)(104,204)(105,202)(106,199)(107,200)(108,201);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1, 
s3*s1*s2*s3*s1*s2*s3*s1*s2*s3*s1*s2, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s3*s2 >;

References

None.

to this polytope.