metabelian, supersoluble, monomial
Aliases: He3.1D9, C27⋊S3⋊2C3, (C3×C27)⋊2C6, C9.1(C9⋊C6), C9○He3.1S3, C9.5He3⋊2C2, C32.2(C3×D9), C9.4(C32⋊C6), C3.3(C32⋊D9), (C3×C9).36(C3×S3), SmallGroup(486,27)
Series: Derived ►Chief ►Lower central ►Upper central
| C3×C27 — He3.D9 |
Generators and relations for He3.D9
G = < a,b,c,d,e | a3=b3=c3=e2=1, d9=ebe=b-1, ab=ba, cac-1=ab-1, ad=da, eae=a-1, bc=cb, bd=db, dcd-1=ece=ab-1c, ede=bd8 >
Character table of He3.D9
| 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 | 6 | 9 | 9 | 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 | 1 | -1 | 1 | 1 | ζ3 | ζ32 | ζ65 | ζ6 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 6 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 3 |
| ρ5 | 1 | -1 | 1 | 1 | ζ32 | ζ3 | ζ6 | ζ65 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | linear of order 6 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | linear of order 3 |
| ρ7 | 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 |
| ρ8 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ10 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ11 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1+√-3 | -1-√-3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | complex lifted from C3×S3 |
| ρ12 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | 2 | 2 | 2 | 2 | 2 | -1-√-3 | -1+√-3 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | complex lifted from C3×S3 |
| ρ13 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | complex lifted from C3×D9 |
| ρ14 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | complex lifted from C3×D9 |
| ρ15 | 2 | 0 | 2 | 2 | -1+√-3 | -1-√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ6 | ζ65 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | complex lifted from C3×D9 |
| ρ16 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ97+ζ95 | ζ98+ζ94 | ζ92+ζ9 | ζ98+ζ97 | ζ94+ζ92 | ζ95+ζ9 | complex lifted from C3×D9 |
| ρ17 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ94 | ζ92+ζ9 | ζ97+ζ95 | ζ94+ζ92 | ζ95+ζ9 | ζ98+ζ97 | complex lifted from C3×D9 |
| ρ18 | 2 | 0 | 2 | 2 | -1-√-3 | -1+√-3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | ζ65 | ζ6 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ92+ζ9 | ζ97+ζ95 | ζ98+ζ94 | ζ95+ζ9 | ζ98+ζ97 | ζ94+ζ92 | complex lifted from C3×D9 |
| ρ19 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | 6 | 6 | 6 | -3 | -3 | 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 | 0 | 0 | 0 | 0 | -3 | -3 | -3 | -3 | 6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ21 | 6 | 0 | 6 | -3 | 0 | 0 | 0 | 0 | -3 | -3 | -3 | 6 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C9⋊C6 |
| ρ22 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | ζ2717+2ζ2710-ζ278+ζ27 | ζ2714+2ζ2713-ζ275+ζ274 | 2ζ2726-ζ2710+ζ278+ζ27 | -ζ2726+2ζ2719+ζ2710+ζ278 | 2ζ2720-ζ2716+ζ277+ζ272 | ζ2725+ζ2711-ζ277+2ζ272 | 2ζ2716+ζ2711+ζ277-ζ272 | 2ζ2723-ζ2713+ζ275+ζ274 | -ζ2723+2ζ2722+ζ2713+ζ275 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ23 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | ζ2725+ζ2711-ζ277+2ζ272 | -ζ2726+2ζ2719+ζ2710+ζ278 | 2ζ2716+ζ2711+ζ277-ζ272 | 2ζ2720-ζ2716+ζ277+ζ272 | 2ζ2723-ζ2713+ζ275+ζ274 | -ζ2723+2ζ2722+ζ2713+ζ275 | ζ2714+2ζ2713-ζ275+ζ274 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2726-ζ2710+ζ278+ζ27 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ24 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | -ζ2723+2ζ2722+ζ2713+ζ275 | 2ζ2720-ζ2716+ζ277+ζ272 | ζ2714+2ζ2713-ζ275+ζ274 | 2ζ2723-ζ2713+ζ275+ζ274 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2726-ζ2710+ζ278+ζ27 | -ζ2726+2ζ2719+ζ2710+ζ278 | ζ2725+ζ2711-ζ277+2ζ272 | 2ζ2716+ζ2711+ζ277-ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ25 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | ζ2714+2ζ2713-ζ275+ζ274 | ζ2725+ζ2711-ζ277+2ζ272 | 2ζ2723-ζ2713+ζ275+ζ274 | -ζ2723+2ζ2722+ζ2713+ζ275 | 2ζ2726-ζ2710+ζ278+ζ27 | -ζ2726+2ζ2719+ζ2710+ζ278 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2716+ζ2711+ζ277-ζ272 | 2ζ2720-ζ2716+ζ277+ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ26 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | -ζ2726+2ζ2719+ζ2710+ζ278 | -ζ2723+2ζ2722+ζ2713+ζ275 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2726-ζ2710+ζ278+ζ27 | 2ζ2716+ζ2711+ζ277-ζ272 | 2ζ2720-ζ2716+ζ277+ζ272 | ζ2725+ζ2711-ζ277+2ζ272 | ζ2714+2ζ2713-ζ275+ζ274 | 2ζ2723-ζ2713+ζ275+ζ274 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ27 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 0 | 0 | 0 | 0 | 2ζ2726-ζ2710+ζ278+ζ27 | 2ζ2723-ζ2713+ζ275+ζ274 | -ζ2726+2ζ2719+ζ2710+ζ278 | ζ2717+2ζ2710-ζ278+ζ27 | ζ2725+ζ2711-ζ277+2ζ272 | 2ζ2716+ζ2711+ζ277-ζ272 | 2ζ2720-ζ2716+ζ277+ζ272 | -ζ2723+2ζ2722+ζ2713+ζ275 | ζ2714+2ζ2713-ζ275+ζ274 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ28 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | 2ζ2720-ζ2716+ζ277+ζ272 | 2ζ2726-ζ2710+ζ278+ζ27 | ζ2725+ζ2711-ζ277+2ζ272 | 2ζ2716+ζ2711+ζ277-ζ272 | ζ2714+2ζ2713-ζ275+ζ274 | 2ζ2723-ζ2713+ζ275+ζ274 | -ζ2723+2ζ2722+ζ2713+ζ275 | -ζ2726+2ζ2719+ζ2710+ζ278 | ζ2717+2ζ2710-ζ278+ζ27 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ29 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 3ζ2715+3ζ2712 | 0 | 0 | 0 | 0 | 2ζ2716+ζ2711+ζ277-ζ272 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2720-ζ2716+ζ277+ζ272 | ζ2725+ζ2711-ζ277+2ζ272 | -ζ2723+2ζ2722+ζ2713+ζ275 | ζ2714+2ζ2713-ζ275+ζ274 | 2ζ2723-ζ2713+ζ275+ζ274 | 2ζ2726-ζ2710+ζ278+ζ27 | -ζ2726+2ζ2719+ζ2710+ζ278 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ30 | 6 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 3ζ2715+3ζ2712 | 3ζ2721+3ζ276 | 3ζ2724+3ζ273 | 0 | 0 | 0 | 0 | 2ζ2723-ζ2713+ζ275+ζ274 | 2ζ2716+ζ2711+ζ277-ζ272 | -ζ2723+2ζ2722+ζ2713+ζ275 | ζ2714+2ζ2713-ζ275+ζ274 | -ζ2726+2ζ2719+ζ2710+ζ278 | ζ2717+2ζ2710-ζ278+ζ27 | 2ζ2726-ζ2710+ζ278+ζ27 | 2ζ2720-ζ2716+ζ277+ζ272 | ζ2725+ζ2711-ζ277+2ζ272 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
►On 81 points
Generators in S81
(1 63 34)(2 64 35)(3 65 36)(4 66 37)(5 67 38)(6 68 39)(7 69 40)(8 70 41)(9 71 42)(10 72 43)(11 73 44)(12 74 45)(13 75 46)(14 76 47)(15 77 48)(16 78 49)(17 79 50)(18 80 51)(19 81 52)(20 55 53)(21 56 54)(22 57 28)(23 58 29)(24 59 30)(25 60 31)(26 61 32)(27 62 33)
(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 64 53)(3 36 74)(5 67 29)(6 39 77)(8 70 32)(9 42 80)(11 73 35)(12 45 56)(14 76 38)(15 48 59)(17 79 41)(18 51 62)(20 55 44)(21 54 65)(23 58 47)(24 30 68)(26 61 50)(27 33 71)(28 37 46)(31 40 49)(34 43 52)(57 75 66)(60 78 69)(63 81 72)
(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 59)(29 58)(30 57)(31 56)(32 55)(33 81)(34 80)(35 79)(36 78)(37 77)(38 76)(39 75)(40 74)(41 73)(42 72)(43 71)(44 70)(45 69)(46 68)(47 67)(48 66)(49 65)(50 64)(51 63)(52 62)(53 61)(54 60)
G:=sub<Sym(81)| (1,63,34)(2,64,35)(3,65,36)(4,66,37)(5,67,38)(6,68,39)(7,69,40)(8,70,41)(9,71,42)(10,72,43)(11,73,44)(12,74,45)(13,75,46)(14,76,47)(15,77,48)(16,78,49)(17,79,50)(18,80,51)(19,81,52)(20,55,53)(21,56,54)(22,57,28)(23,58,29)(24,59,30)(25,60,31)(26,61,32)(27,62,33), (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), (2,64,53)(3,36,74)(5,67,29)(6,39,77)(8,70,32)(9,42,80)(11,73,35)(12,45,56)(14,76,38)(15,48,59)(17,79,41)(18,51,62)(20,55,44)(21,54,65)(23,58,47)(24,30,68)(26,61,50)(27,33,71)(28,37,46)(31,40,49)(34,43,52)(57,75,66)(60,78,69)(63,81,72), (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)(28,59)(29,58)(30,57)(31,56)(32,55)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60)>;
G:=Group( (1,63,34)(2,64,35)(3,65,36)(4,66,37)(5,67,38)(6,68,39)(7,69,40)(8,70,41)(9,71,42)(10,72,43)(11,73,44)(12,74,45)(13,75,46)(14,76,47)(15,77,48)(16,78,49)(17,79,50)(18,80,51)(19,81,52)(20,55,53)(21,56,54)(22,57,28)(23,58,29)(24,59,30)(25,60,31)(26,61,32)(27,62,33), (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), (2,64,53)(3,36,74)(5,67,29)(6,39,77)(8,70,32)(9,42,80)(11,73,35)(12,45,56)(14,76,38)(15,48,59)(17,79,41)(18,51,62)(20,55,44)(21,54,65)(23,58,47)(24,30,68)(26,61,50)(27,33,71)(28,37,46)(31,40,49)(34,43,52)(57,75,66)(60,78,69)(63,81,72), (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)(28,59)(29,58)(30,57)(31,56)(32,55)(33,81)(34,80)(35,79)(36,78)(37,77)(38,76)(39,75)(40,74)(41,73)(42,72)(43,71)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,64)(51,63)(52,62)(53,61)(54,60) );
G=PermutationGroup([[(1,63,34),(2,64,35),(3,65,36),(4,66,37),(5,67,38),(6,68,39),(7,69,40),(8,70,41),(9,71,42),(10,72,43),(11,73,44),(12,74,45),(13,75,46),(14,76,47),(15,77,48),(16,78,49),(17,79,50),(18,80,51),(19,81,52),(20,55,53),(21,56,54),(22,57,28),(23,58,29),(24,59,30),(25,60,31),(26,61,32),(27,62,33)], [(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)], [(2,64,53),(3,36,74),(5,67,29),(6,39,77),(8,70,32),(9,42,80),(11,73,35),(12,45,56),(14,76,38),(15,48,59),(17,79,41),(18,51,62),(20,55,44),(21,54,65),(23,58,47),(24,30,68),(26,61,50),(27,33,71),(28,37,46),(31,40,49),(34,43,52),(57,75,66),(60,78,69),(63,81,72)], [(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),(28,59),(29,58),(30,57),(31,56),(32,55),(33,81),(34,80),(35,79),(36,78),(37,77),(38,76),(39,75),(40,74),(41,73),(42,72),(43,71),(44,70),(45,69),(46,68),(47,67),(48,66),(49,65),(50,64),(51,63),(52,62),(53,61),(54,60)]])
Matrix representation of He3.D9 ►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 | 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 |
| 1 | 8 | 8 | 100 | 1 | 8 |
| 101 | 9 | 9 | 108 | 101 | 9 |
| 1 | 8 | 1 | 8 | 8 | 100 |
| 101 | 9 | 101 | 9 | 9 | 108 |
| 8 | 100 | 1 | 8 | 1 | 8 |
| 9 | 108 | 101 | 9 | 101 | 9 |
| 9 | 108 | 108 | 101 | 9 | 108 |
| 8 | 100 | 100 | 1 | 8 | 100 |
| 108 | 101 | 9 | 108 | 9 | 108 |
| 100 | 1 | 8 | 100 | 8 | 100 |
| 9 | 108 | 9 | 108 | 108 | 101 |
| 8 | 100 | 8 | 100 | 100 | 1 |
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],[108,108,0,0,0,0,1,0,0,0,0,0,0,0,108,108,0,0,0,0,1,0,0,0,0,0,0,0,108,108,0,0,0,0,1,0],[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],[1,101,1,101,8,9,8,9,8,9,100,108,8,9,1,101,1,101,100,108,8,9,8,9,1,101,8,9,1,101,8,9,100,108,8,9],[9,8,108,100,9,8,108,100,101,1,108,100,108,100,9,8,9,8,101,1,108,100,108,100,9,8,9,8,108,100,108,100,108,100,101,1] >;
He3.D9 in GAP, Magma, Sage, TeX
{\rm He}_3.D_9 % in TeX
G:=Group("He3.D9"); // GroupNames label
G:=SmallGroup(486,27);
// by ID
G=gap.SmallGroup(486,27);
# by ID
G:=PCGroup([6,-2,-3,-3,-3,-3,-3,1190,224,1310,867,2169,12964,118,11669]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^3=c^3=e^2=1,d^9=e*b*e=b^-1,a*b=b*a,c*a*c^-1=a*b^-1,a*d=d*a,e*a*e=a^-1,b*c=c*b,b*d=d*b,d*c*d^-1=e*c*e=a*b^-1*c,e*d*e=b*d^8>;
// generators/relations
Export
Subgroup lattice of He3.D9 in TeX
Character table of He3.D9 in TeX