metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: Dic27, C27⋊C4, C54.C2, C2.D27, C6.1D9, C9.Dic3, C3.Dic9, C18.1S3, SmallGroup(108,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — Dic27 |
Generators and relations for Dic27
G = < a,b | a54=1, b2=a27, bab-1=a-1 >
Character table of Dic27
| class | 1 | 2 | 3 | 4A | 4B | 6 | 9A | 9B | 9C | 18A | 18B | 18C | 27A | 27B | 27C | 27D | 27E | 27F | 27G | 27H | 27I | 54A | 54B | 54C | 54D | 54E | 54F | 54G | 54H | 54I | |
| size | 1 | 1 | 2 | 27 | 27 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | |
| 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 | i | -i | -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 4 |
| ρ4 | 1 | -1 | 1 | -i | i | -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 4 |
| ρ5 | 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 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ6 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ7 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | orthogonal lifted from D27 |
| ρ8 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 2 | 0 | 0 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | orthogonal lifted from D27 |
| ρ11 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | orthogonal lifted from D27 |
| ρ12 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | orthogonal lifted from D27 |
| ρ13 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | orthogonal lifted from D27 |
| ρ14 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | orthogonal lifted from D27 |
| ρ15 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | orthogonal lifted from D27 |
| ρ16 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | orthogonal lifted from D27 |
| ρ17 | 2 | 2 | -1 | 0 | 0 | -1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | orthogonal lifted from D27 |
| ρ18 | 2 | -2 | 2 | 0 | 0 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | symplectic lifted from Dic9, Schur index 2 |
| ρ19 | 2 | -2 | 2 | 0 | 0 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | symplectic lifted from Dic9, Schur index 2 |
| ρ20 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | -ζ2721-ζ276 | -ζ2724-ζ273 | -ζ2715-ζ2712 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | -ζ2726-ζ27 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2717-ζ2710 | -ζ2719-ζ278 | symplectic faithful, Schur index 2 |
| ρ21 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | -ζ2724-ζ273 | -ζ2715-ζ2712 | -ζ2721-ζ276 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | -ζ2722-ζ275 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2723-ζ274 | -ζ2714-ζ2713 | symplectic faithful, Schur index 2 |
| ρ22 | 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 | 1 | 1 | 1 | symplectic lifted from Dic3, Schur index 2 |
| ρ23 | 2 | -2 | 2 | 0 | 0 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | symplectic lifted from Dic9, Schur index 2 |
| ρ24 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | -ζ2715-ζ2712 | -ζ2721-ζ276 | -ζ2724-ζ273 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | -ζ2725-ζ272 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2720-ζ277 | -ζ2716-ζ2711 | symplectic faithful, Schur index 2 |
| ρ25 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | -ζ2724-ζ273 | -ζ2715-ζ2712 | -ζ2721-ζ276 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | -ζ2714-ζ2713 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2722-ζ275 | -ζ2723-ζ274 | symplectic faithful, Schur index 2 |
| ρ26 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | -ζ2721-ζ276 | -ζ2724-ζ273 | -ζ2715-ζ2712 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | -ζ2717-ζ2710 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2719-ζ278 | -ζ2726-ζ27 | symplectic faithful, Schur index 2 |
| ρ27 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2715+ζ2712 | ζ2721+ζ276 | ζ2724+ζ273 | -ζ2724-ζ273 | -ζ2715-ζ2712 | -ζ2721-ζ276 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | -ζ2723-ζ274 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2714-ζ2713 | -ζ2722-ζ275 | symplectic faithful, Schur index 2 |
| ρ28 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | -ζ2715-ζ2712 | -ζ2721-ζ276 | -ζ2724-ζ273 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | -ζ2716-ζ2711 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2725-ζ272 | -ζ2720-ζ277 | symplectic faithful, Schur index 2 |
| ρ29 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2724+ζ273 | ζ2715+ζ2712 | ζ2721+ζ276 | -ζ2721-ζ276 | -ζ2724-ζ273 | -ζ2715-ζ2712 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | -ζ2719-ζ278 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2726-ζ27 | -ζ2717-ζ2710 | symplectic faithful, Schur index 2 |
| ρ30 | 2 | -2 | -1 | 0 | 0 | 1 | ζ2721+ζ276 | ζ2724+ζ273 | ζ2715+ζ2712 | -ζ2715-ζ2712 | -ζ2721-ζ276 | -ζ2724-ζ273 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | -ζ2720-ζ277 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2716-ζ2711 | -ζ2725-ζ272 | symplectic faithful, Schur index 2 |
►Regular action on 108 points
Generators in S108
(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)
(1 60 28 87)(2 59 29 86)(3 58 30 85)(4 57 31 84)(5 56 32 83)(6 55 33 82)(7 108 34 81)(8 107 35 80)(9 106 36 79)(10 105 37 78)(11 104 38 77)(12 103 39 76)(13 102 40 75)(14 101 41 74)(15 100 42 73)(16 99 43 72)(17 98 44 71)(18 97 45 70)(19 96 46 69)(20 95 47 68)(21 94 48 67)(22 93 49 66)(23 92 50 65)(24 91 51 64)(25 90 52 63)(26 89 53 62)(27 88 54 61)
G:=sub<Sym(108)| (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), (1,60,28,87)(2,59,29,86)(3,58,30,85)(4,57,31,84)(5,56,32,83)(6,55,33,82)(7,108,34,81)(8,107,35,80)(9,106,36,79)(10,105,37,78)(11,104,38,77)(12,103,39,76)(13,102,40,75)(14,101,41,74)(15,100,42,73)(16,99,43,72)(17,98,44,71)(18,97,45,70)(19,96,46,69)(20,95,47,68)(21,94,48,67)(22,93,49,66)(23,92,50,65)(24,91,51,64)(25,90,52,63)(26,89,53,62)(27,88,54,61)>;
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)(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108), (1,60,28,87)(2,59,29,86)(3,58,30,85)(4,57,31,84)(5,56,32,83)(6,55,33,82)(7,108,34,81)(8,107,35,80)(9,106,36,79)(10,105,37,78)(11,104,38,77)(12,103,39,76)(13,102,40,75)(14,101,41,74)(15,100,42,73)(16,99,43,72)(17,98,44,71)(18,97,45,70)(19,96,46,69)(20,95,47,68)(21,94,48,67)(22,93,49,66)(23,92,50,65)(24,91,51,64)(25,90,52,63)(26,89,53,62)(27,88,54,61) );
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),(55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108)], [(1,60,28,87),(2,59,29,86),(3,58,30,85),(4,57,31,84),(5,56,32,83),(6,55,33,82),(7,108,34,81),(8,107,35,80),(9,106,36,79),(10,105,37,78),(11,104,38,77),(12,103,39,76),(13,102,40,75),(14,101,41,74),(15,100,42,73),(16,99,43,72),(17,98,44,71),(18,97,45,70),(19,96,46,69),(20,95,47,68),(21,94,48,67),(22,93,49,66),(23,92,50,65),(24,91,51,64),(25,90,52,63),(26,89,53,62),(27,88,54,61)]])
Dic27 is a maximal subgroup of
Dic54 C4×D27 C27⋊D4 Dic81 C27⋊C12 C27⋊Dic3 Q8.D27 C18.S4
Dic27 is a maximal quotient of
C27⋊C8 Dic81 C27⋊Dic3 C18.S4
Matrix representation of Dic27 ►in GL2(𝔽109) generated by
| 16 | 79 |
| 30 | 46 |
| 65 | 67 |
| 2 | 44 |
G:=sub<GL(2,GF(109))| [16,30,79,46],[65,2,67,44] >;
Dic27 in GAP, Magma, Sage, TeX
{\rm Dic}_{27} % in TeX
G:=Group("Dic27"); // GroupNames label
G:=SmallGroup(108,1);
// by ID
G=gap.SmallGroup(108,1);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,10,302,237,1203,138,1804]);
// Polycyclic
G:=Group<a,b|a^54=1,b^2=a^27,b*a*b^-1=a^-1>;
// generators/relations
Export
Subgroup lattice of Dic27 in TeX
Character table of Dic27 in TeX