metabelian, supersoluble, monomial
Aliases: (C32×C9)⋊14C6, He3⋊C3⋊3S3, C32⋊4D9⋊6C3, He3.2(C3⋊S3), C3⋊(He3.2S3), (C3×He3).17S3, C33.66(C3×S3), C3.8(He3⋊4S3), C32.19(C32⋊C6), (C3×C9)⋊19(C3×S3), (C3×He3⋊C3)⋊6C2, C32.18(C3×C3⋊S3), SmallGroup(486,173)
Series: Derived ►Chief ►Lower central ►Upper central
| C32×C9 — He3⋊C3⋊3S3 |
Generators and relations for He3⋊C3⋊3S3
G = < a,b,c,d,e,f | a3=b3=c3=d3=e3=f2=1, ab=ba, cac-1=dad-1=ab-1, ae=ea, faf=a-1, fdf=bc=cb, fcf=bd=db, be=eb, fbf=b-1, dcd-1=abc, ce=ec, de=ed, fef=e-1 >
Subgroups: 1196 in 102 conjugacy classes, 22 normal (11 characteristic)
C1, C2, C3, C3, C3, S3, C6, C9, C32, C32, C32, D9, C3×S3, C3⋊S3, C3×C9, C3×C9, He3, He3, C33, C33, C32⋊C6, C9⋊S3, C3×C3⋊S3, C33⋊C2, He3⋊C3, He3⋊C3, C32×C9, C3×He3, C3×He3, He3.2S3, He3⋊4S3, C32⋊4D9, C3×He3⋊C3, He3⋊C3⋊3S3
Quotients: C1, C2, C3, S3, C6, C3×S3, C3⋊S3, C32⋊C6, C3×C3⋊S3, He3.2S3, He3⋊4S3, He3⋊C3⋊3S3
Character table of He3⋊C3⋊3S3
| class | 1 | 2 | 3A | 3B | 3C | 3D | 3E | 3F | 3G | 3H | 3I | 3J | 3K | 3L | 3M | 3N | 3O | 3P | 3Q | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 9H | 9I | |
| size | 1 | 81 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 9 | 9 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 81 | 81 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ6 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 6 |
| ρ7 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | orthogonal lifted from S3 |
| ρ8 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | -1 | -1 | -1 | 2 | -1 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | -1 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ10 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | 2 | -1 | -1 | 2 | -1 | 2 | -1 | -1 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | orthogonal lifted from S3 |
| ρ11 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1-√-3 | ζ65 | ζ65 | -1+√-3 | ζ6 | ζ6 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | ζ65 | ζ65 | ζ6 | -1-√-3 | ζ6 | -1+√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | -1-√-3 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | complex lifted from C3×S3 |
| ρ14 | 2 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1+√-3 | ζ6 | ζ6 | -1-√-3 | ζ65 | ζ65 | ζ65 | ζ6 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ15 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ65 | ζ65 | -1+√-3 | ζ6 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ16 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | -1+√-3 | ζ65 | ζ65 | ζ6 | ζ6 | -1-√-3 | ζ65 | 0 | 0 | -1 | -1 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | complex lifted from C3×S3 |
| ρ17 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1+√-3 | -1-√-3 | ζ6 | ζ65 | -1+√-3 | ζ65 | -1-√-3 | ζ6 | ζ6 | ζ65 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | 2 | -1 | complex lifted from C3×S3 |
| ρ18 | 2 | 0 | 2 | -1 | -1 | -1 | 2 | -1 | -1 | -1-√-3 | -1+√-3 | ζ65 | ζ6 | ζ6 | ζ6 | ζ65 | -1+√-3 | ζ65 | -1-√-3 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | complex lifted from C3×S3 |
| ρ19 | 6 | 0 | 6 | 6 | 6 | 6 | -3 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ20 | 6 | 0 | 6 | -3 | -3 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ21 | 6 | 0 | 6 | -3 | -3 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C32⋊C6 |
| ρ22 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3.2S3 |
| ρ23 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ24 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3.2S3 |
| ρ25 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ26 | 6 | 0 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ27 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | orthogonal lifted from He3.2S3 |
| ρ28 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ29 | 6 | 0 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | 2ζ98-ζ94+ζ92+ζ9 | orthogonal lifted from He3.2S3 |
| ρ30 | 6 | 0 | -3 | 6 | -3 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | 2ζ98-ζ94+ζ92+ζ9 | -ζ98+2ζ97+ζ94+ζ92 | ζ95+2ζ94-ζ92+ζ9 | orthogonal lifted from He3.2S3 |
►On 81 points
Generators in S81
(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)
(1 14 10)(2 15 11)(3 13 12)(4 80 8)(5 81 9)(6 79 7)(16 23 19)(17 24 20)(18 22 21)(25 31 29)(26 32 30)(27 33 28)(34 39 41)(35 37 42)(36 38 40)(43 48 50)(44 46 51)(45 47 49)(52 55 58)(53 56 59)(54 57 60)(61 67 64)(62 68 65)(63 69 66)(70 76 73)(71 77 74)(72 78 75)
(1 4 47)(2 81 43)(3 7 51)(5 50 11)(6 44 13)(8 45 10)(9 48 15)(12 79 46)(14 80 49)(16 62 57)(17 69 52)(18 64 59)(19 65 54)(20 63 58)(21 67 56)(22 61 53)(23 68 60)(24 66 55)(25 71 40)(26 78 39)(27 73 35)(28 76 42)(29 74 38)(30 72 34)(31 77 36)(32 75 41)(33 70 37)
(1 70 54)(2 77 58)(3 75 56)(4 39 24)(5 42 18)(6 36 19)(7 40 23)(8 34 17)(9 37 21)(10 73 60)(11 71 55)(12 78 53)(13 72 59)(14 76 57)(15 74 52)(16 79 38)(20 80 41)(22 81 35)(25 67 47)(26 65 43)(27 63 51)(28 66 46)(29 61 45)(30 68 50)(31 64 49)(32 62 48)(33 69 44)
(1 3 2)(4 7 81)(5 8 79)(6 9 80)(10 12 11)(13 15 14)(16 18 17)(19 21 20)(22 24 23)(25 27 26)(28 30 29)(31 33 32)(34 38 42)(35 39 40)(36 37 41)(43 47 51)(44 48 49)(45 46 50)(52 57 59)(53 55 60)(54 56 58)(61 66 68)(62 64 69)(63 65 67)(70 75 77)(71 73 78)(72 74 76)
(2 3)(4 76)(5 78)(6 77)(7 74)(8 73)(9 75)(10 14)(11 13)(12 15)(16 29)(17 28)(18 30)(19 25)(20 27)(21 26)(22 32)(23 31)(24 33)(34 37)(35 39)(36 38)(41 42)(43 53)(44 52)(45 54)(46 58)(47 60)(48 59)(49 57)(50 56)(51 55)(61 68)(62 67)(63 69)(64 65)(70 80)(71 79)(72 81)
G:=sub<Sym(81)| (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), (1,14,10)(2,15,11)(3,13,12)(4,80,8)(5,81,9)(6,79,7)(16,23,19)(17,24,20)(18,22,21)(25,31,29)(26,32,30)(27,33,28)(34,39,41)(35,37,42)(36,38,40)(43,48,50)(44,46,51)(45,47,49)(52,55,58)(53,56,59)(54,57,60)(61,67,64)(62,68,65)(63,69,66)(70,76,73)(71,77,74)(72,78,75), (1,4,47)(2,81,43)(3,7,51)(5,50,11)(6,44,13)(8,45,10)(9,48,15)(12,79,46)(14,80,49)(16,62,57)(17,69,52)(18,64,59)(19,65,54)(20,63,58)(21,67,56)(22,61,53)(23,68,60)(24,66,55)(25,71,40)(26,78,39)(27,73,35)(28,76,42)(29,74,38)(30,72,34)(31,77,36)(32,75,41)(33,70,37), (1,70,54)(2,77,58)(3,75,56)(4,39,24)(5,42,18)(6,36,19)(7,40,23)(8,34,17)(9,37,21)(10,73,60)(11,71,55)(12,78,53)(13,72,59)(14,76,57)(15,74,52)(16,79,38)(20,80,41)(22,81,35)(25,67,47)(26,65,43)(27,63,51)(28,66,46)(29,61,45)(30,68,50)(31,64,49)(32,62,48)(33,69,44), (1,3,2)(4,7,81)(5,8,79)(6,9,80)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,38,42)(35,39,40)(36,37,41)(43,47,51)(44,48,49)(45,46,50)(52,57,59)(53,55,60)(54,56,58)(61,66,68)(62,64,69)(63,65,67)(70,75,77)(71,73,78)(72,74,76), (2,3)(4,76)(5,78)(6,77)(7,74)(8,73)(9,75)(10,14)(11,13)(12,15)(16,29)(17,28)(18,30)(19,25)(20,27)(21,26)(22,32)(23,31)(24,33)(34,37)(35,39)(36,38)(41,42)(43,53)(44,52)(45,54)(46,58)(47,60)(48,59)(49,57)(50,56)(51,55)(61,68)(62,67)(63,69)(64,65)(70,80)(71,79)(72,81)>;
G:=Group( (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), (1,14,10)(2,15,11)(3,13,12)(4,80,8)(5,81,9)(6,79,7)(16,23,19)(17,24,20)(18,22,21)(25,31,29)(26,32,30)(27,33,28)(34,39,41)(35,37,42)(36,38,40)(43,48,50)(44,46,51)(45,47,49)(52,55,58)(53,56,59)(54,57,60)(61,67,64)(62,68,65)(63,69,66)(70,76,73)(71,77,74)(72,78,75), (1,4,47)(2,81,43)(3,7,51)(5,50,11)(6,44,13)(8,45,10)(9,48,15)(12,79,46)(14,80,49)(16,62,57)(17,69,52)(18,64,59)(19,65,54)(20,63,58)(21,67,56)(22,61,53)(23,68,60)(24,66,55)(25,71,40)(26,78,39)(27,73,35)(28,76,42)(29,74,38)(30,72,34)(31,77,36)(32,75,41)(33,70,37), (1,70,54)(2,77,58)(3,75,56)(4,39,24)(5,42,18)(6,36,19)(7,40,23)(8,34,17)(9,37,21)(10,73,60)(11,71,55)(12,78,53)(13,72,59)(14,76,57)(15,74,52)(16,79,38)(20,80,41)(22,81,35)(25,67,47)(26,65,43)(27,63,51)(28,66,46)(29,61,45)(30,68,50)(31,64,49)(32,62,48)(33,69,44), (1,3,2)(4,7,81)(5,8,79)(6,9,80)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,38,42)(35,39,40)(36,37,41)(43,47,51)(44,48,49)(45,46,50)(52,57,59)(53,55,60)(54,56,58)(61,66,68)(62,64,69)(63,65,67)(70,75,77)(71,73,78)(72,74,76), (2,3)(4,76)(5,78)(6,77)(7,74)(8,73)(9,75)(10,14)(11,13)(12,15)(16,29)(17,28)(18,30)(19,25)(20,27)(21,26)(22,32)(23,31)(24,33)(34,37)(35,39)(36,38)(41,42)(43,53)(44,52)(45,54)(46,58)(47,60)(48,59)(49,57)(50,56)(51,55)(61,68)(62,67)(63,69)(64,65)(70,80)(71,79)(72,81) );
G=PermutationGroup([[(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)], [(1,14,10),(2,15,11),(3,13,12),(4,80,8),(5,81,9),(6,79,7),(16,23,19),(17,24,20),(18,22,21),(25,31,29),(26,32,30),(27,33,28),(34,39,41),(35,37,42),(36,38,40),(43,48,50),(44,46,51),(45,47,49),(52,55,58),(53,56,59),(54,57,60),(61,67,64),(62,68,65),(63,69,66),(70,76,73),(71,77,74),(72,78,75)], [(1,4,47),(2,81,43),(3,7,51),(5,50,11),(6,44,13),(8,45,10),(9,48,15),(12,79,46),(14,80,49),(16,62,57),(17,69,52),(18,64,59),(19,65,54),(20,63,58),(21,67,56),(22,61,53),(23,68,60),(24,66,55),(25,71,40),(26,78,39),(27,73,35),(28,76,42),(29,74,38),(30,72,34),(31,77,36),(32,75,41),(33,70,37)], [(1,70,54),(2,77,58),(3,75,56),(4,39,24),(5,42,18),(6,36,19),(7,40,23),(8,34,17),(9,37,21),(10,73,60),(11,71,55),(12,78,53),(13,72,59),(14,76,57),(15,74,52),(16,79,38),(20,80,41),(22,81,35),(25,67,47),(26,65,43),(27,63,51),(28,66,46),(29,61,45),(30,68,50),(31,64,49),(32,62,48),(33,69,44)], [(1,3,2),(4,7,81),(5,8,79),(6,9,80),(10,12,11),(13,15,14),(16,18,17),(19,21,20),(22,24,23),(25,27,26),(28,30,29),(31,33,32),(34,38,42),(35,39,40),(36,37,41),(43,47,51),(44,48,49),(45,46,50),(52,57,59),(53,55,60),(54,56,58),(61,66,68),(62,64,69),(63,65,67),(70,75,77),(71,73,78),(72,74,76)], [(2,3),(4,76),(5,78),(6,77),(7,74),(8,73),(9,75),(10,14),(11,13),(12,15),(16,29),(17,28),(18,30),(19,25),(20,27),(21,26),(22,32),(23,31),(24,33),(34,37),(35,39),(36,38),(41,42),(43,53),(44,52),(45,54),(46,58),(47,60),(48,59),(49,57),(50,56),(51,55),(61,68),(62,67),(63,69),(64,65),(70,80),(71,79),(72,81)]])
Matrix representation of He3⋊C3⋊3S3 ►in GL8(𝔽19)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 11 | 0 | 0 | 7 | 1 | 0 |
| 0 | 0 | 11 | 0 | 0 | 7 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 18 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 18 | 0 | 0 | 0 |
| 0 | 0 | 11 | 0 | 7 | 0 | 0 | 1 |
| 0 | 0 | 0 | 8 | 0 | 12 | 18 | 18 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 14 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 2 | 0 | 0 |
| 0 | 0 | 18 | 18 | 8 | 8 | 7 | 10 |
| 0 | 0 | 2 | 2 | 3 | 3 | 16 | 7 |
| 0 | 0 | 0 | 5 | 0 | 17 | 16 | 11 |
| 0 | 0 | 7 | 0 | 0 | 17 | 16 | 11 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 17 | 0 | 0 |
| 0 | 0 | 16 | 16 | 5 | 5 | 3 | 10 |
| 0 | 0 | 17 | 17 | 16 | 16 | 12 | 3 |
| 0 | 0 | 14 | 2 | 2 | 3 | 3 | 14 |
| 0 | 0 | 0 | 14 | 2 | 3 | 3 | 14 |
| 18 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 18 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 18 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 18 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 18 | 0 | 0 |
| 0 | 0 | 8 | 0 | 12 | 0 | 18 | 18 |
| 0 | 0 | 0 | 11 | 0 | 7 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 8 | 8 | 12 | 12 | 18 | 18 |
G:=sub<GL(8,GF(19))| [1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,0,0,11,11,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,18,18,7,7,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,18,18,0,0,11,0,0,0,1,0,0,0,0,8,0,0,0,0,18,18,7,0,0,0,0,0,1,0,0,12,0,0,0,0,0,0,0,18,0,0,0,0,0,0,1,18],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,18,2,0,7,0,0,0,0,18,2,5,0,0,0,7,5,8,3,0,0,0,0,14,2,8,3,17,17,0,0,0,0,7,16,16,16,0,0,0,0,10,7,11,11],[1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,16,17,14,0,0,0,0,0,16,17,2,14,0,0,5,7,5,16,2,2,0,0,12,17,5,16,3,3,0,0,0,0,3,12,3,3,0,0,0,0,10,3,14,14],[18,18,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,8,0,0,0,18,18,0,0,0,11,0,0,0,0,0,1,12,0,0,0,0,0,18,18,0,7,0,0,0,0,0,0,18,1,0,0,0,0,0,0,18,0],[0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,8,0,0,1,0,0,0,0,8,0,0,0,0,0,1,0,12,0,0,0,0,1,0,0,12,0,0,0,0,0,0,1,18,0,0,0,0,0,0,0,18] >;
He3⋊C3⋊3S3 in GAP, Magma, Sage, TeX
{\rm He}_3\rtimes C_3\rtimes_3S_3 % in TeX
G:=Group("He3:C3:3S3"); // GroupNames label
G:=SmallGroup(486,173);
// by ID
G=gap.SmallGroup(486,173);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,3134,548,986,867,3244,3250,11669]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^3=c^3=d^3=e^3=f^2=1,a*b=b*a,c*a*c^-1=d*a*d^-1=a*b^-1,a*e=e*a,f*a*f=a^-1,f*d*f=b*c=c*b,f*c*f=b*d=d*b,b*e=e*b,f*b*f=b^-1,d*c*d^-1=a*b*c,c*e=e*c,d*e=e*d,f*e*f=e^-1>;
// generators/relations
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