Polytope of Type {6,12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12,4}*1728s
if this polytope has a name.
Group : SmallGroup(1728,47870)
Rank : 4
Schlafli Type : {6,12,4}
Number of vertices, edges, etc : 18, 108, 72, 4
Order of s₀s₁s₂s₃ : 12
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12,4}*864d
   9-fold quotients : {2,12,4}*192b
   18-fold quotients : {2,6,4}*96c
   36-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope