direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D54, C2×D27, C54⋊C2, C9.D6, C27⋊C22, C3.D18, C6.2D9, C18.2S3, sometimes denoted D108 or Dih54 or Dih108, SmallGroup(108,4)
Series: Derived ►Chief ►Lower central ►Upper central
| C27 — D54 |
Generators and relations for D54
G = < a,b | a54=b2=1, bab=a-1 >
Character table of D54
| class | 1 | 2A | 2B | 2C | 3 | 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 | 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 | 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 | -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 |
| ρ4 | 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 |
| ρ5 | 2 | 2 | 0 | 0 | 2 | 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 | 0 | 0 | 2 | -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 D6 |
| ρ7 | 2 | -2 | 0 | 0 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | orthogonal lifted from D18 |
| ρ8 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | orthogonal lifted from D9 |
| ρ9 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | orthogonal lifted from D9 |
| ρ10 | 2 | 2 | 0 | 0 | 2 | 2 | -1 | -1 | -1 | -1 | -1 | -1 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | orthogonal lifted from D9 |
| ρ11 | 2 | -2 | 0 | 0 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | orthogonal lifted from D18 |
| ρ12 | 2 | -2 | 0 | 0 | 2 | -2 | -1 | -1 | -1 | 1 | 1 | 1 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -ζ97-ζ92 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ95-ζ94 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ98-ζ9 | -ζ97-ζ92 | -ζ97-ζ92 | orthogonal lifted from D18 |
| ρ13 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2716+ζ2711 | -ζ2717-ζ2710 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2719-ζ278 | -ζ2726-ζ27 | orthogonal faithful |
| ρ14 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | orthogonal lifted from D27 |
| ρ15 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2714+ζ2713 | -ζ2725-ζ272 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2720-ζ277 | -ζ2716-ζ2711 | orthogonal faithful |
| ρ16 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | orthogonal lifted from D27 |
| ρ17 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2725+ζ272 | -ζ2719-ζ278 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2726-ζ27 | -ζ2717-ζ2710 | orthogonal faithful |
| ρ18 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | orthogonal lifted from D27 |
| ρ19 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2720+ζ277 | -ζ2726-ζ27 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2717-ζ2710 | -ζ2719-ζ278 | orthogonal faithful |
| ρ20 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2719+ζ278 | -ζ2722-ζ275 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2723-ζ274 | -ζ2714-ζ2713 | orthogonal faithful |
| ρ21 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | orthogonal lifted from D27 |
| ρ22 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2717+ζ2710 | -ζ2714-ζ2713 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2716-ζ2711 | -ζ2722-ζ275 | -ζ2723-ζ274 | orthogonal faithful |
| ρ23 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | orthogonal lifted from D27 |
| ρ24 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2723+ζ274 | -ζ2716-ζ2711 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2722-ζ275 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2726-ζ27 | -ζ2725-ζ272 | -ζ2720-ζ277 | orthogonal faithful |
| ρ25 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | -ζ2724-ζ273 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2726+ζ27 | -ζ2723-ζ274 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2716-ζ2711 | -ζ2725-ζ272 | -ζ2720-ζ277 | -ζ2714-ζ2713 | -ζ2722-ζ275 | orthogonal faithful |
| ρ26 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2716+ζ2711 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | orthogonal lifted from D27 |
| ρ27 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2714+ζ2713 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2723+ζ274 | orthogonal lifted from D27 |
| ρ28 | 2 | -2 | 0 | 0 | -1 | 1 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | -ζ2724-ζ273 | -ζ2721-ζ276 | -ζ2715-ζ2712 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2717+ζ2710 | ζ2720+ζ277 | ζ2722+ζ275 | -ζ2720-ζ277 | -ζ2722-ζ275 | -ζ2723-ζ274 | -ζ2714-ζ2713 | -ζ2726-ζ27 | -ζ2717-ζ2710 | -ζ2719-ζ278 | -ζ2716-ζ2711 | -ζ2725-ζ272 | orthogonal faithful |
| ρ29 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2716+ζ2711 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2719+ζ278 | orthogonal lifted from D27 |
| ρ30 | 2 | 2 | 0 | 0 | -1 | -1 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2715+ζ2712 | ζ2724+ζ273 | ζ2721+ζ276 | ζ2725+ζ272 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2717+ζ2710 | ζ2716+ζ2711 | ζ2725+ζ272 | ζ2720+ζ277 | ζ2714+ζ2713 | ζ2722+ζ275 | ζ2723+ζ274 | ζ2719+ζ278 | ζ2726+ζ27 | orthogonal lifted from D27 |
►On 54 points
Generators in S54
(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)
(1 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 48)(8 47)(9 46)(10 45)(11 44)(12 43)(13 42)(14 41)(15 40)(16 39)(17 38)(18 37)(19 36)(20 35)(21 34)(22 33)(23 32)(24 31)(25 30)(26 29)(27 28)
G:=sub<Sym(54)| (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)>;
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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,48)(8,47)(9,46)(10,45)(11,44)(12,43)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28) );
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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,48),(8,47),(9,46),(10,45),(11,44),(12,43),(13,42),(14,41),(15,40),(16,39),(17,38),(18,37),(19,36),(20,35),(21,34),(22,33),(23,32),(24,31),(25,30),(26,29),(27,28)]])
D54 is a maximal subgroup of
D108 C27⋊D4 Q8⋊D27
D54 is a maximal quotient of Dic54 D108 C27⋊D4
Matrix representation of D54 ►in GL2(𝔽109) generated by
| 22 | 51 |
| 58 | 80 |
| 22 | 51 |
| 29 | 87 |
G:=sub<GL(2,GF(109))| [22,58,51,80],[22,29,51,87] >;
D54 in GAP, Magma, Sage, TeX
D_{54} % in TeX
G:=Group("D54"); // GroupNames label
G:=SmallGroup(108,4);
// by ID
G=gap.SmallGroup(108,4);
# by ID
G:=PCGroup([5,-2,-2,-3,-3,-3,302,237,1203,138,1804]);
// Polycyclic
G:=Group<a,b|a^54=b^2=1,b*a*b=a^-1>;
// generators/relations
Export
Subgroup lattice of D54 in TeX
Character table of D54 in TeX