metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C68.1C4, D34.4C4, C17⋊1M4(2), Dic17.5C22, C4.(C17⋊C4), C17⋊2C8⋊1C2, C34.2(C2×C4), (C4×D17).3C2, C2.4(C2×C17⋊C4), SmallGroup(272,30)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D34.4C4
G = < a,b | a68=1, b4=a34, bab-1=a55 >
Character table of D34.4C4
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | 68B | 68C | 68D | 68E | 68F | 68G | 68H | |
| size | 1 | 1 | 34 | 2 | 17 | 17 | 34 | 34 | 34 | 34 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | |
| 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 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | -1 | -1 | -1 | i | -i | -i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | -1 | -1 | -1 | -i | i | i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 2 | -2 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ12 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | orthogonal lifted from C2×C17⋊C4 |
| ρ13 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ14 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ15 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | orthogonal lifted from C2×C17⋊C4 |
| ρ16 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | orthogonal lifted from C2×C17⋊C4 |
| ρ17 | 4 | 4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | orthogonal lifted from C2×C17⋊C4 |
| ρ18 | 4 | 4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ19 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | complex faithful |
| ρ20 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | complex faithful |
| ρ21 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | complex faithful |
| ρ22 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | complex faithful |
| ρ23 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | complex faithful |
| ρ24 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | complex faithful |
| ρ25 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | complex faithful |
| ρ26 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | ζ43ζ1711-ζ43ζ1710-ζ43ζ177+ζ43ζ176 | ζ4ζ1711-ζ4ζ1710-ζ4ζ177+ζ4ζ176 | ζ4ζ1715-ζ4ζ179-ζ4ζ178+ζ4ζ172 | ζ43ζ1716-ζ43ζ1713-ζ43ζ174+ζ43ζ17 | ζ43ζ1715-ζ43ζ179-ζ43ζ178+ζ43ζ172 | -ζ4ζ1714+ζ4ζ1712+ζ4ζ175-ζ4ζ173 | -ζ43ζ1716+ζ43ζ1713+ζ43ζ174-ζ43ζ17 | ζ4ζ1714-ζ4ζ1712-ζ4ζ175+ζ4ζ173 | complex faithful |
►On 136 points
Generators in S136
(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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136)
(1 86 52 103 35 120 18 69)(2 133 17 90 36 99 51 124)(3 112 50 77 37 78 16 111)(4 91 15 132 38 125 49 98)(5 70 48 119 39 104 14 85)(6 117 13 106 40 83 47 72)(7 96 46 93 41 130 12 127)(8 75 11 80 42 109 45 114)(9 122 44 135 43 88 10 101)(19 116 34 73 53 82 68 107)(20 95 67 128 54 129 33 94)(21 74 32 115 55 108 66 81)(22 121 65 102 56 87 31 136)(23 100 30 89 57 134 64 123)(24 79 63 76 58 113 29 110)(25 126 28 131 59 92 62 97)(26 105 61 118 60 71 27 84)
G:=sub<Sym(136)| (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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,86,52,103,35,120,18,69)(2,133,17,90,36,99,51,124)(3,112,50,77,37,78,16,111)(4,91,15,132,38,125,49,98)(5,70,48,119,39,104,14,85)(6,117,13,106,40,83,47,72)(7,96,46,93,41,130,12,127)(8,75,11,80,42,109,45,114)(9,122,44,135,43,88,10,101)(19,116,34,73,53,82,68,107)(20,95,67,128,54,129,33,94)(21,74,32,115,55,108,66,81)(22,121,65,102,56,87,31,136)(23,100,30,89,57,134,64,123)(24,79,63,76,58,113,29,110)(25,126,28,131,59,92,62,97)(26,105,61,118,60,71,27,84)>;
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)(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136), (1,86,52,103,35,120,18,69)(2,133,17,90,36,99,51,124)(3,112,50,77,37,78,16,111)(4,91,15,132,38,125,49,98)(5,70,48,119,39,104,14,85)(6,117,13,106,40,83,47,72)(7,96,46,93,41,130,12,127)(8,75,11,80,42,109,45,114)(9,122,44,135,43,88,10,101)(19,116,34,73,53,82,68,107)(20,95,67,128,54,129,33,94)(21,74,32,115,55,108,66,81)(22,121,65,102,56,87,31,136)(23,100,30,89,57,134,64,123)(24,79,63,76,58,113,29,110)(25,126,28,131,59,92,62,97)(26,105,61,118,60,71,27,84) );
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),(69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136)], [(1,86,52,103,35,120,18,69),(2,133,17,90,36,99,51,124),(3,112,50,77,37,78,16,111),(4,91,15,132,38,125,49,98),(5,70,48,119,39,104,14,85),(6,117,13,106,40,83,47,72),(7,96,46,93,41,130,12,127),(8,75,11,80,42,109,45,114),(9,122,44,135,43,88,10,101),(19,116,34,73,53,82,68,107),(20,95,67,128,54,129,33,94),(21,74,32,115,55,108,66,81),(22,121,65,102,56,87,31,136),(23,100,30,89,57,134,64,123),(24,79,63,76,58,113,29,110),(25,126,28,131,59,92,62,97),(26,105,61,118,60,71,27,84)]])
Matrix representation of D34.4C4 ►in GL6(𝔽137)
| 37 | 0 | 0 | 0 | 0 | 0 |
| 133 | 100 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 12 | 13 | 86 |
| 0 | 0 | 51 | 89 | 39 | 101 |
| 0 | 0 | 36 | 97 | 100 | 85 |
| 0 | 0 | 52 | 72 | 52 | 136 |
| 129 | 126 | 0 | 0 | 0 | 0 |
| 59 | 8 | 0 | 0 | 0 | 0 |
| 0 | 0 | 61 | 20 | 18 | 26 |
| 0 | 0 | 76 | 47 | 26 | 41 |
| 0 | 0 | 70 | 61 | 96 | 24 |
| 0 | 0 | 24 | 96 | 61 | 70 |
G:=sub<GL(6,GF(137))| [37,133,0,0,0,0,0,100,0,0,0,0,0,0,1,51,36,52,0,0,12,89,97,72,0,0,13,39,100,52,0,0,86,101,85,136],[129,59,0,0,0,0,126,8,0,0,0,0,0,0,61,76,70,24,0,0,20,47,61,96,0,0,18,26,96,61,0,0,26,41,24,70] >;
D34.4C4 in GAP, Magma, Sage, TeX
D_{34}._4C_4 % in TeX
G:=Group("D34.4C4"); // GroupNames label
G:=SmallGroup(272,30);
// by ID
G=gap.SmallGroup(272,30);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,46,42,5204,1614]);
// Polycyclic
G:=Group<a,b|a^68=1,b^4=a^34,b*a*b^-1=a^55>;
// generators/relations
Export
Subgroup lattice of D34.4C4 in TeX
Character table of D34.4C4 in TeX