metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C17⋊2M4(2), Dic17.4C4, Dic17.7C22, C17⋊2C8⋊2C2, C34.6(C2×C4), (C2×C34).2C4, C22.(C17⋊C4), (C2×Dic17).5C2, C2.6(C2×C17⋊C4), SmallGroup(272,34)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C17⋊M4(2)
G = < a,b,c | a17=b8=c2=1, bab-1=a4, ac=ca, cbc=b5 >
Character table of C17⋊M4(2)
| class | 1 | 2A | 2B | 4A | 4B | 4C | 8A | 8B | 8C | 8D | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | 34E | 34F | 34G | 34H | 34I | 34J | 34K | 34L | |
| size | 1 | 1 | 2 | 17 | 17 | 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 | -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 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | complex lifted from M4(2) |
| ρ10 | 2 | -2 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | complex lifted from M4(2) |
| ρ11 | 4 | 4 | 4 | 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 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ12 | 4 | 4 | -4 | 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 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | orthogonal lifted from C2×C17⋊C4 |
| ρ13 | 4 | 4 | 4 | 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 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ14 | 4 | 4 | -4 | 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 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | orthogonal lifted from C2×C17⋊C4 |
| ρ15 | 4 | 4 | 4 | 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 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ16 | 4 | 4 | -4 | 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 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1715-ζ179-ζ178-ζ172 | orthogonal lifted from C2×C17⋊C4 |
| ρ17 | 4 | 4 | 4 | 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 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ18 | 4 | 4 | -4 | 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 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | ζ1715+ζ179+ζ178+ζ172 | 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 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715+ζ179+ζ178-ζ172 | -ζ1714+ζ1712+ζ175-ζ173 | ζ1715-ζ179-ζ178+ζ172 | ζ1716-ζ1713-ζ174+ζ17 | -ζ1716+ζ1713+ζ174-ζ17 | -ζ1711+ζ1710+ζ177-ζ176 | ζ1714-ζ1712-ζ175+ζ173 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | ζ1711-ζ1710-ζ177+ζ176 | symplectic faithful, Schur index 2 |
| 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 | -ζ1716+ζ1713+ζ174-ζ17 | ζ1711-ζ1710-ζ177+ζ176 | ζ1716-ζ1713-ζ174+ζ17 | -ζ1715+ζ179+ζ178-ζ172 | ζ1715-ζ179-ζ178+ζ172 | -ζ1714+ζ1712+ζ175-ζ173 | -ζ1711+ζ1710+ζ177-ζ176 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | ζ1714-ζ1712-ζ175+ζ173 | symplectic faithful, Schur index 2 |
| ρ21 | 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 | ζ1711-ζ1710-ζ177+ζ176 | -ζ1715+ζ179+ζ178-ζ172 | -ζ1711+ζ1710+ζ177-ζ176 | -ζ1714+ζ1712+ζ175-ζ173 | ζ1714-ζ1712-ζ175+ζ173 | ζ1716-ζ1713-ζ174+ζ17 | ζ1715-ζ179-ζ178+ζ172 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716+ζ1713+ζ174-ζ17 | symplectic faithful, Schur index 2 |
| ρ22 | 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 | ζ1714-ζ1712-ζ175+ζ173 | -ζ1716+ζ1713+ζ174-ζ17 | -ζ1714+ζ1712+ζ175-ζ173 | ζ1711-ζ1710-ζ177+ζ176 | -ζ1711+ζ1710+ζ177-ζ176 | -ζ1715+ζ179+ζ178-ζ172 | ζ1716-ζ1713-ζ174+ζ17 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | ζ1715-ζ179-ζ178+ζ172 | symplectic faithful, Schur index 2 |
| ρ23 | 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 | -ζ1711+ζ1710+ζ177-ζ176 | ζ1715-ζ179-ζ178+ζ172 | ζ1711-ζ1710-ζ177+ζ176 | ζ1714-ζ1712-ζ175+ζ173 | -ζ1714+ζ1712+ζ175-ζ173 | -ζ1716+ζ1713+ζ174-ζ17 | -ζ1715+ζ179+ζ178-ζ172 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | ζ1716-ζ1713-ζ174+ζ17 | symplectic faithful, Schur index 2 |
| ρ24 | 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 | ζ1715-ζ179-ζ178+ζ172 | ζ1714-ζ1712-ζ175+ζ173 | -ζ1715+ζ179+ζ178-ζ172 | -ζ1716+ζ1713+ζ174-ζ17 | ζ1716-ζ1713-ζ174+ζ17 | ζ1711-ζ1710-ζ177+ζ176 | -ζ1714+ζ1712+ζ175-ζ173 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711+ζ1710+ζ177-ζ176 | symplectic faithful, Schur index 2 |
| 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 | -ζ1714+ζ1712+ζ175-ζ173 | ζ1716-ζ1713-ζ174+ζ17 | ζ1714-ζ1712-ζ175+ζ173 | -ζ1711+ζ1710+ζ177-ζ176 | ζ1711-ζ1710-ζ177+ζ176 | ζ1715-ζ179-ζ178+ζ172 | -ζ1716+ζ1713+ζ174-ζ17 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715+ζ179+ζ178-ζ172 | symplectic faithful, Schur index 2 |
| ρ26 | 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 | ζ1716-ζ1713-ζ174+ζ17 | -ζ1711+ζ1710+ζ177-ζ176 | -ζ1716+ζ1713+ζ174-ζ17 | ζ1715-ζ179-ζ178+ζ172 | -ζ1715+ζ179+ζ178-ζ172 | ζ1714-ζ1712-ζ175+ζ173 | ζ1711-ζ1710-ζ177+ζ176 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714+ζ1712+ζ175-ζ173 | symplectic faithful, Schur index 2 |
►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)
(1 108 35 100 18 130 52 70)(2 104 51 87 19 126 68 74)(3 117 50 91 20 122 67 78)(4 113 49 95 21 135 66 82)(5 109 48 99 22 131 65 69)(6 105 47 86 23 127 64 73)(7 118 46 90 24 123 63 77)(8 114 45 94 25 136 62 81)(9 110 44 98 26 132 61 85)(10 106 43 102 27 128 60 72)(11 119 42 89 28 124 59 76)(12 115 41 93 29 120 58 80)(13 111 40 97 30 133 57 84)(14 107 39 101 31 129 56 71)(15 103 38 88 32 125 55 75)(16 116 37 92 33 121 54 79)(17 112 36 96 34 134 53 83)
(69 99)(70 100)(71 101)(72 102)(73 86)(74 87)(75 88)(76 89)(77 90)(78 91)(79 92)(80 93)(81 94)(82 95)(83 96)(84 97)(85 98)(103 125)(104 126)(105 127)(106 128)(107 129)(108 130)(109 131)(110 132)(111 133)(112 134)(113 135)(114 136)(115 120)(116 121)(117 122)(118 123)(119 124)
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), (1,108,35,100,18,130,52,70)(2,104,51,87,19,126,68,74)(3,117,50,91,20,122,67,78)(4,113,49,95,21,135,66,82)(5,109,48,99,22,131,65,69)(6,105,47,86,23,127,64,73)(7,118,46,90,24,123,63,77)(8,114,45,94,25,136,62,81)(9,110,44,98,26,132,61,85)(10,106,43,102,27,128,60,72)(11,119,42,89,28,124,59,76)(12,115,41,93,29,120,58,80)(13,111,40,97,30,133,57,84)(14,107,39,101,31,129,56,71)(15,103,38,88,32,125,55,75)(16,116,37,92,33,121,54,79)(17,112,36,96,34,134,53,83), (69,99)(70,100)(71,101)(72,102)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,120)(116,121)(117,122)(118,123)(119,124)>;
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), (1,108,35,100,18,130,52,70)(2,104,51,87,19,126,68,74)(3,117,50,91,20,122,67,78)(4,113,49,95,21,135,66,82)(5,109,48,99,22,131,65,69)(6,105,47,86,23,127,64,73)(7,118,46,90,24,123,63,77)(8,114,45,94,25,136,62,81)(9,110,44,98,26,132,61,85)(10,106,43,102,27,128,60,72)(11,119,42,89,28,124,59,76)(12,115,41,93,29,120,58,80)(13,111,40,97,30,133,57,84)(14,107,39,101,31,129,56,71)(15,103,38,88,32,125,55,75)(16,116,37,92,33,121,54,79)(17,112,36,96,34,134,53,83), (69,99)(70,100)(71,101)(72,102)(73,86)(74,87)(75,88)(76,89)(77,90)(78,91)(79,92)(80,93)(81,94)(82,95)(83,96)(84,97)(85,98)(103,125)(104,126)(105,127)(106,128)(107,129)(108,130)(109,131)(110,132)(111,133)(112,134)(113,135)(114,136)(115,120)(116,121)(117,122)(118,123)(119,124) );
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)], [(1,108,35,100,18,130,52,70),(2,104,51,87,19,126,68,74),(3,117,50,91,20,122,67,78),(4,113,49,95,21,135,66,82),(5,109,48,99,22,131,65,69),(6,105,47,86,23,127,64,73),(7,118,46,90,24,123,63,77),(8,114,45,94,25,136,62,81),(9,110,44,98,26,132,61,85),(10,106,43,102,27,128,60,72),(11,119,42,89,28,124,59,76),(12,115,41,93,29,120,58,80),(13,111,40,97,30,133,57,84),(14,107,39,101,31,129,56,71),(15,103,38,88,32,125,55,75),(16,116,37,92,33,121,54,79),(17,112,36,96,34,134,53,83)], [(69,99),(70,100),(71,101),(72,102),(73,86),(74,87),(75,88),(76,89),(77,90),(78,91),(79,92),(80,93),(81,94),(82,95),(83,96),(84,97),(85,98),(103,125),(104,126),(105,127),(106,128),(107,129),(108,130),(109,131),(110,132),(111,133),(112,134),(113,135),(114,136),(115,120),(116,121),(117,122),(118,123),(119,124)]])
Matrix representation of C17⋊M4(2) ►in GL4(𝔽137) generated by
| 136 | 1 | 0 | 0 |
| 11 | 125 | 0 | 0 |
| 0 | 0 | 128 | 116 |
| 0 | 0 | 43 | 85 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 118 | 70 | 0 | 0 |
| 124 | 19 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 136 |
G:=sub<GL(4,GF(137))| [136,11,0,0,1,125,0,0,0,0,128,43,0,0,116,85],[0,0,118,124,0,0,70,19,1,0,0,0,0,1,0,0],[1,0,0,0,0,1,0,0,0,0,136,0,0,0,0,136] >;
C17⋊M4(2) in GAP, Magma, Sage, TeX
C_{17}\rtimes M_4(2) % in TeX
G:=Group("C17:M4(2)"); // GroupNames label
G:=SmallGroup(272,34);
// by ID
G=gap.SmallGroup(272,34);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,42,5204,1614]);
// Polycyclic
G:=Group<a,b,c|a^17=b^8=c^2=1,b*a*b^-1=a^4,a*c=c*a,c*b*c=b^5>;
// generators/relations
Export
Subgroup lattice of C17⋊M4(2) in TeX
Character table of C17⋊M4(2) in TeX