metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C68⋊1C4, D17.Q8, D17.1D4, Dic17⋊3C4, D34.5C22, C17⋊(C4⋊C4), C4⋊(C17⋊C4), C34.4(C2×C4), (C4×D17).4C2, C2.5(C2×C17⋊C4), (C2×C17⋊C4).1C2, SmallGroup(272,32)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C68⋊C4
G = < a,b | a68=b4=1, bab-1=a55 >
Character table of C68⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 4E | 4F | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 68A | 68B | 68C | 68D | 68E | 68F | 68G | 68H | |
| size | 1 | 1 | 17 | 17 | 2 | 34 | 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 | i | i | -i | -i | 1 | 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 | -i | i | -i | i | -1 | 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 | -i | -i | i | i | 1 | 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 | i | -i | i | -i | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 2 | -2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 4 | 4 | 0 | 0 | 4 | 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 | 0 | 4 | 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 |
| ρ13 | 4 | 4 | 0 | 0 | 4 | 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 |
| ρ14 | 4 | 4 | 0 | 0 | -4 | 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 |
| ρ15 | 4 | 4 | 0 | 0 | -4 | 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 |
| ρ16 | 4 | 4 | 0 | 0 | -4 | 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 |
| ρ17 | 4 | 4 | 0 | 0 | 4 | 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 |
| ρ18 | 4 | 4 | 0 | 0 | -4 | 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 |
| 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 68 points
Generators in S68
(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)
(1 52)(2 31 34 39)(3 10 67 26)(4 57 32 13)(5 36 65 68)(6 15 30 55)(7 62 63 42)(8 41 28 29)(9 20 61 16)(11 46 59 58)(12 25 24 45)(14 51 22 19)(17 56 53 48)(18 35)(21 40 49 64)(23 66 47 38)(27 50 43 54)(33 60 37 44)
G:=sub<Sym(68)| (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), (1,52)(2,31,34,39)(3,10,67,26)(4,57,32,13)(5,36,65,68)(6,15,30,55)(7,62,63,42)(8,41,28,29)(9,20,61,16)(11,46,59,58)(12,25,24,45)(14,51,22,19)(17,56,53,48)(18,35)(21,40,49,64)(23,66,47,38)(27,50,43,54)(33,60,37,44)>;
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), (1,52)(2,31,34,39)(3,10,67,26)(4,57,32,13)(5,36,65,68)(6,15,30,55)(7,62,63,42)(8,41,28,29)(9,20,61,16)(11,46,59,58)(12,25,24,45)(14,51,22,19)(17,56,53,48)(18,35)(21,40,49,64)(23,66,47,38)(27,50,43,54)(33,60,37,44) );
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)], [(1,52),(2,31,34,39),(3,10,67,26),(4,57,32,13),(5,36,65,68),(6,15,30,55),(7,62,63,42),(8,41,28,29),(9,20,61,16),(11,46,59,58),(12,25,24,45),(14,51,22,19),(17,56,53,48),(18,35),(21,40,49,64),(23,66,47,38),(27,50,43,54),(33,60,37,44)]])
Matrix representation of C68⋊C4 ►in GL4(𝔽137) generated by
| 53 | 113 | 122 | 99 |
| 77 | 48 | 73 | 107 |
| 116 | 39 | 106 | 116 |
| 101 | 99 | 113 | 28 |
| 37 | 114 | 61 | 42 |
| 53 | 27 | 8 | 76 |
| 111 | 27 | 111 | 110 |
| 43 | 7 | 13 | 99 |
G:=sub<GL(4,GF(137))| [53,77,116,101,113,48,39,99,122,73,106,113,99,107,116,28],[37,53,111,43,114,27,27,7,61,8,111,13,42,76,110,99] >;
C68⋊C4 in GAP, Magma, Sage, TeX
C_{68}\rtimes C_4 % in TeX
G:=Group("C68:C4"); // GroupNames label
G:=SmallGroup(272,32);
// by ID
G=gap.SmallGroup(272,32);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,46,5204,1614]);
// Polycyclic
G:=Group<a,b|a^68=b^4=1,b*a*b^-1=a^55>;
// generators/relations
Export
Subgroup lattice of C68⋊C4 in TeX
Character table of C68⋊C4 in TeX