non-abelian, supersoluble, monomial
Aliases: He3.4D9, 3- 1+2.2D9, C27⋊C3⋊3S3, (C3×C27)⋊6S3, C9○He3.4S3, C9.6He3⋊1C2, C32.4(C9⋊S3), C9.4(He3⋊C2), C3.10(C32⋊2D9), (C3×C9).11(C3⋊S3), SmallGroup(486,59)
Series: Derived ►Chief ►Lower central ►Upper central
| C9.6He3 — He3.4D9 |
Generators and relations for He3.4D9
G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=ebe=b-1, dad-1=eae=ab=ba, cac-1=ab-1, bc=cb, bd=db, dcd-1=a-1c, ece=abc-1, ede=bd8 >
Character table of He3.4D9
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 9G | 27A | 27B | 27C | 27D | 27E | 27F | 27G | 27H | 27I | 27J | 27K | 27L | 27M | 27N | 27O | |
| size | 1 | 81 | 2 | 3 | 3 | 18 | 81 | 81 | 2 | 2 | 2 | 6 | 6 | 18 | 18 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 18 | 18 | |
| 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 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ4 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | 2 | -1 | 2 | -1 | orthogonal lifted from S3 |
| ρ5 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | -1 | 2 | -1 | 2 | orthogonal lifted from S3 |
| ρ7 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | 2 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ95+ζ94 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ12 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ13 | 2 | 0 | 2 | 2 | 2 | -1 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | 2 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ14 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ15 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ16 | 3 | -1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | ζ65 | ζ6 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ17 | 3 | -1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | ζ6 | ζ65 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ18 | 3 | 1 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | ζ32 | ζ3 | 3 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ19 | 3 | 1 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | ζ3 | ζ32 | 3 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from He3⋊C2 |
| ρ20 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2723+ζ2722-ζ2713+2ζ275 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2726+ζ2710-ζ278+2ζ27 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2723+ζ2713-ζ275+2ζ274 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ21 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2716+2ζ2711-ζ277+ζ272 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2717+ζ2710+ζ278-ζ27 | 2ζ2725-ζ2711+ζ277+ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2726+ζ2710-ζ278+2ζ27 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2723+ζ2722-ζ2713+2ζ275 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ24 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2720+ζ2716+2ζ277-ζ272 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2726+ζ2710-ζ278+2ζ27 | ζ2716+2ζ2711-ζ277+ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ25 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2717+ζ2710+ζ278-ζ27 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2726+ζ2719-ζ2710+2ζ278 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ27 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2723+ζ2713-ζ275+2ζ274 | 2ζ2714+ζ2713+ζ275-ζ274 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2717+ζ2710+ζ278-ζ27 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2720+ζ2716+2ζ277-ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ28 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2721+3ζ276 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | ζ2723+ζ2722-ζ2713+2ζ275 | ζ2723+ζ2713-ζ275+2ζ274 | ζ2726+ζ2719-ζ2710+2ζ278 | ζ2726+ζ2710-ζ278+2ζ27 | 2ζ2717+ζ2710+ζ278-ζ27 | 2ζ2725-ζ2711+ζ277+ζ272 | ζ2720+ζ2716+2ζ277-ζ272 | ζ2716+2ζ2711-ζ277+ζ272 | 2ζ2714+ζ2713+ζ275-ζ274 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ29 | 6 | 0 | 6 | -3-3√-3 | -3+3√-3 | 0 | 0 | 0 | -3 | -3 | -3 | 3+3√-3/2 | 3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊2D9 |
| ρ30 | 6 | 0 | 6 | -3+3√-3 | -3-3√-3 | 0 | 0 | 0 | -3 | -3 | -3 | 3-3√-3/2 | 3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C32⋊2D9 |
►On 81 points
Generators in S81
(1 80 49)(2 72 32)(3 64 42)(4 56 52)(5 75 35)(6 67 45)(7 59 28)(8 78 38)(9 70 48)(10 62 31)(11 81 41)(12 73 51)(13 65 34)(14 57 44)(15 76 54)(16 68 37)(17 60 47)(18 79 30)(19 71 40)(20 63 50)(21 55 33)(22 74 43)(23 66 53)(24 58 36)(25 77 46)(26 69 29)(27 61 39)
(1 19 10)(2 20 11)(3 21 12)(4 22 13)(5 23 14)(6 24 15)(7 25 16)(8 26 17)(9 27 18)(28 46 37)(29 47 38)(30 48 39)(31 49 40)(32 50 41)(33 51 42)(34 52 43)(35 53 44)(36 54 45)(55 73 64)(56 74 65)(57 75 66)(58 76 67)(59 77 68)(60 78 69)(61 79 70)(62 80 71)(63 81 72)
(2 32 81)(3 55 42)(5 35 57)(6 58 45)(8 38 60)(9 61 48)(11 41 63)(12 64 51)(14 44 66)(15 67 54)(17 47 69)(18 70 30)(20 50 72)(21 73 33)(23 53 75)(24 76 36)(26 29 78)(27 79 39)(28 37 46)(31 40 49)(34 43 52)(56 74 65)(59 77 68)(62 80 71)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 13)(7 12)(8 11)(9 10)(19 27)(20 26)(21 25)(22 24)(28 33)(29 32)(30 31)(34 54)(35 53)(36 52)(37 51)(38 50)(39 49)(40 48)(41 47)(42 46)(43 45)(55 68)(56 67)(57 66)(58 65)(59 64)(60 63)(61 62)(69 81)(70 80)(71 79)(72 78)(73 77)(74 76)
G:=sub<Sym(81)| (1,80,49)(2,72,32)(3,64,42)(4,56,52)(5,75,35)(6,67,45)(7,59,28)(8,78,38)(9,70,48)(10,62,31)(11,81,41)(12,73,51)(13,65,34)(14,57,44)(15,76,54)(16,68,37)(17,60,47)(18,79,30)(19,71,40)(20,63,50)(21,55,33)(22,74,43)(23,66,53)(24,58,36)(25,77,46)(26,69,29)(27,61,39), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45)(55,73,64)(56,74,65)(57,75,66)(58,76,67)(59,77,68)(60,78,69)(61,79,70)(62,80,71)(63,81,72), (2,32,81)(3,55,42)(5,35,57)(6,58,45)(8,38,60)(9,61,48)(11,41,63)(12,64,51)(14,44,66)(15,67,54)(17,47,69)(18,70,30)(20,50,72)(21,73,33)(23,53,75)(24,76,36)(26,29,78)(27,79,39)(28,37,46)(31,40,49)(34,43,52)(56,74,65)(59,77,68)(62,80,71), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,27)(20,26)(21,25)(22,24)(28,33)(29,32)(30,31)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76)>;
G:=Group( (1,80,49)(2,72,32)(3,64,42)(4,56,52)(5,75,35)(6,67,45)(7,59,28)(8,78,38)(9,70,48)(10,62,31)(11,81,41)(12,73,51)(13,65,34)(14,57,44)(15,76,54)(16,68,37)(17,60,47)(18,79,30)(19,71,40)(20,63,50)(21,55,33)(22,74,43)(23,66,53)(24,58,36)(25,77,46)(26,69,29)(27,61,39), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18)(28,46,37)(29,47,38)(30,48,39)(31,49,40)(32,50,41)(33,51,42)(34,52,43)(35,53,44)(36,54,45)(55,73,64)(56,74,65)(57,75,66)(58,76,67)(59,77,68)(60,78,69)(61,79,70)(62,80,71)(63,81,72), (2,32,81)(3,55,42)(5,35,57)(6,58,45)(8,38,60)(9,61,48)(11,41,63)(12,64,51)(14,44,66)(15,67,54)(17,47,69)(18,70,30)(20,50,72)(21,73,33)(23,53,75)(24,76,36)(26,29,78)(27,79,39)(28,37,46)(31,40,49)(34,43,52)(56,74,65)(59,77,68)(62,80,71), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)(19,27)(20,26)(21,25)(22,24)(28,33)(29,32)(30,31)(34,54)(35,53)(36,52)(37,51)(38,50)(39,49)(40,48)(41,47)(42,46)(43,45)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)(69,81)(70,80)(71,79)(72,78)(73,77)(74,76) );
G=PermutationGroup([[(1,80,49),(2,72,32),(3,64,42),(4,56,52),(5,75,35),(6,67,45),(7,59,28),(8,78,38),(9,70,48),(10,62,31),(11,81,41),(12,73,51),(13,65,34),(14,57,44),(15,76,54),(16,68,37),(17,60,47),(18,79,30),(19,71,40),(20,63,50),(21,55,33),(22,74,43),(23,66,53),(24,58,36),(25,77,46),(26,69,29),(27,61,39)], [(1,19,10),(2,20,11),(3,21,12),(4,22,13),(5,23,14),(6,24,15),(7,25,16),(8,26,17),(9,27,18),(28,46,37),(29,47,38),(30,48,39),(31,49,40),(32,50,41),(33,51,42),(34,52,43),(35,53,44),(36,54,45),(55,73,64),(56,74,65),(57,75,66),(58,76,67),(59,77,68),(60,78,69),(61,79,70),(62,80,71),(63,81,72)], [(2,32,81),(3,55,42),(5,35,57),(6,58,45),(8,38,60),(9,61,48),(11,41,63),(12,64,51),(14,44,66),(15,67,54),(17,47,69),(18,70,30),(20,50,72),(21,73,33),(23,53,75),(24,76,36),(26,29,78),(27,79,39),(28,37,46),(31,40,49),(34,43,52),(56,74,65),(59,77,68),(62,80,71)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,13),(7,12),(8,11),(9,10),(19,27),(20,26),(21,25),(22,24),(28,33),(29,32),(30,31),(34,54),(35,53),(36,52),(37,51),(38,50),(39,49),(40,48),(41,47),(42,46),(43,45),(55,68),(56,67),(57,66),(58,65),(59,64),(60,63),(61,62),(69,81),(70,80),(71,79),(72,78),(73,77),(74,76)]])
Matrix representation of He3.4D9 ►in GL6(𝔽109)
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 108 | 108 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 108 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 108 | 108 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 108 | 108 | 0 | 0 |
| 0 | 0 | 0 | 0 | 108 | 108 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 98 | 41 | 52 | 11 | 98 | 41 |
| 68 | 57 | 98 | 41 | 68 | 57 |
| 52 | 11 | 52 | 11 | 68 | 57 |
| 98 | 41 | 98 | 41 | 52 | 11 |
| 98 | 41 | 68 | 57 | 68 | 57 |
| 68 | 57 | 52 | 11 | 52 | 11 |
| 68 | 57 | 98 | 41 | 68 | 57 |
| 98 | 41 | 52 | 11 | 98 | 41 |
| 98 | 41 | 98 | 41 | 52 | 11 |
| 52 | 11 | 52 | 11 | 68 | 57 |
| 68 | 57 | 52 | 11 | 52 | 11 |
| 98 | 41 | 68 | 57 | 68 | 57 |
G:=sub<GL(6,GF(109))| [0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0],[108,1,0,0,0,0,108,0,0,0,0,0,0,0,108,1,0,0,0,0,108,0,0,0,0,0,0,0,108,1,0,0,0,0,108,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,108,0,0,0,0,1,108,0,0,0,0,0,0,108,1,0,0,0,0,108,0],[98,68,52,98,98,68,41,57,11,41,41,57,52,98,52,98,68,52,11,41,11,41,57,11,98,68,68,52,68,52,41,57,57,11,57,11],[68,98,98,52,68,98,57,41,41,11,57,41,98,52,98,52,52,68,41,11,41,11,11,57,68,98,52,68,52,68,57,41,11,57,11,57] >;
He3.4D9 in GAP, Magma, Sage, TeX
{\rm He}_3._4D_9 % in TeX
G:=Group("He3.4D9"); // GroupNames label
G:=SmallGroup(486,59);
// by ID
G=gap.SmallGroup(486,59);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,265,1195,218,548,4755,453,3250,1906,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^9=e*b*e=b^-1,d*a*d^-1=e*a*e=a*b=b*a,c*a*c^-1=a*b^-1,b*c=c*b,b*d=d*b,d*c*d^-1=a^-1*c,e*c*e=a*b*c^-1,e*d*e=b*d^8>;
// generators/relations
Export
Subgroup lattice of He3.4D9 in TeX
Character table of He3.4D9 in TeX