metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D34⋊2C4, D17.2D4, D34.6C22, (C2×C34)⋊1C4, C17⋊(C22⋊C4), C22⋊(C17⋊C4), C34.7(C2×C4), (C22×D17).2C2, (C2×C17⋊C4)⋊C2, C2.7(C2×C17⋊C4), SmallGroup(272,35)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D17.D4
G = < a,b,c,d | a17=b2=c4=1, d2=a-1b, bab=a-1, cac-1=dad-1=a4, cbc-1=dbd-1=a3b, dcd-1=a-1bc-1 >
2C2
17C2
17C2
34C2
17C22
17C22
34C4
34C4
34C22
34C22
2D17
2C34
17C23
17C2×C4
17C2×C4
2D34
2D34
17C22⋊C4
Character table of D17.D4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 4A | 4B | 4C | 4D | 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 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | orthogonal lifted from D4 |
| ρ11 | 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 | orthogonal faithful |
| ρ12 | 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 | orthogonal faithful |
| ρ13 | 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 | orthogonal faithful |
| 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 | 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 |
| ρ16 | 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 |
| ρ17 | 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 |
| 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 | 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 |
| ρ20 | 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 |
| ρ21 | 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 | orthogonal faithful |
| 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 | orthogonal faithful |
| 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 | orthogonal faithful |
| 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 | orthogonal faithful |
| ρ25 | 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 |
| 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 | orthogonal 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 28)(2 27)(3 26)(4 25)(5 24)(6 23)(7 22)(8 21)(9 20)(10 19)(11 18)(12 34)(13 33)(14 32)(15 31)(16 30)(17 29)(35 67)(36 66)(37 65)(38 64)(39 63)(40 62)(41 61)(42 60)(43 59)(44 58)(45 57)(46 56)(47 55)(48 54)(49 53)(50 52)(51 68)
(1 44)(2 40 17 48)(3 36 16 35)(4 49 15 39)(5 45 14 43)(6 41 13 47)(7 37 12 51)(8 50 11 38)(9 46 10 42)(18 52 23 66)(19 65 22 53)(20 61 21 57)(24 62 34 56)(25 58 33 60)(26 54 32 64)(27 67 31 68)(28 63 30 55)(29 59)
(1 59 29 44)(2 55 28 48)(3 68 27 35)(4 64 26 39)(5 60 25 43)(6 56 24 47)(7 52 23 51)(8 65 22 38)(9 61 21 42)(10 57 20 46)(11 53 19 50)(12 66 18 37)(13 62 34 41)(14 58 33 45)(15 54 32 49)(16 67 31 36)(17 63 30 40)
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(51,68), (1,44)(2,40,17,48)(3,36,16,35)(4,49,15,39)(5,45,14,43)(6,41,13,47)(7,37,12,51)(8,50,11,38)(9,46,10,42)(18,52,23,66)(19,65,22,53)(20,61,21,57)(24,62,34,56)(25,58,33,60)(26,54,32,64)(27,67,31,68)(28,63,30,55)(29,59), (1,59,29,44)(2,55,28,48)(3,68,27,35)(4,64,26,39)(5,60,25,43)(6,56,24,47)(7,52,23,51)(8,65,22,38)(9,61,21,42)(10,57,20,46)(11,53,19,50)(12,66,18,37)(13,62,34,41)(14,58,33,45)(15,54,32,49)(16,67,31,36)(17,63,30,40)>;
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), (1,28)(2,27)(3,26)(4,25)(5,24)(6,23)(7,22)(8,21)(9,20)(10,19)(11,18)(12,34)(13,33)(14,32)(15,31)(16,30)(17,29)(35,67)(36,66)(37,65)(38,64)(39,63)(40,62)(41,61)(42,60)(43,59)(44,58)(45,57)(46,56)(47,55)(48,54)(49,53)(50,52)(51,68), (1,44)(2,40,17,48)(3,36,16,35)(4,49,15,39)(5,45,14,43)(6,41,13,47)(7,37,12,51)(8,50,11,38)(9,46,10,42)(18,52,23,66)(19,65,22,53)(20,61,21,57)(24,62,34,56)(25,58,33,60)(26,54,32,64)(27,67,31,68)(28,63,30,55)(29,59), (1,59,29,44)(2,55,28,48)(3,68,27,35)(4,64,26,39)(5,60,25,43)(6,56,24,47)(7,52,23,51)(8,65,22,38)(9,61,21,42)(10,57,20,46)(11,53,19,50)(12,66,18,37)(13,62,34,41)(14,58,33,45)(15,54,32,49)(16,67,31,36)(17,63,30,40) );
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)], [(1,28),(2,27),(3,26),(4,25),(5,24),(6,23),(7,22),(8,21),(9,20),(10,19),(11,18),(12,34),(13,33),(14,32),(15,31),(16,30),(17,29),(35,67),(36,66),(37,65),(38,64),(39,63),(40,62),(41,61),(42,60),(43,59),(44,58),(45,57),(46,56),(47,55),(48,54),(49,53),(50,52),(51,68)], [(1,44),(2,40,17,48),(3,36,16,35),(4,49,15,39),(5,45,14,43),(6,41,13,47),(7,37,12,51),(8,50,11,38),(9,46,10,42),(18,52,23,66),(19,65,22,53),(20,61,21,57),(24,62,34,56),(25,58,33,60),(26,54,32,64),(27,67,31,68),(28,63,30,55),(29,59)], [(1,59,29,44),(2,55,28,48),(3,68,27,35),(4,64,26,39),(5,60,25,43),(6,56,24,47),(7,52,23,51),(8,65,22,38),(9,61,21,42),(10,57,20,46),(11,53,19,50),(12,66,18,37),(13,62,34,41),(14,58,33,45),(15,54,32,49),(16,67,31,36),(17,63,30,40)]])
Matrix representation of D17.D4 ►in GL6(𝔽137)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 103 | 1 | 0 | 0 |
| 0 | 0 | 56 | 0 | 1 | 0 |
| 0 | 0 | 111 | 0 | 0 | 1 |
| 0 | 0 | 117 | 57 | 106 | 86 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 107 | 121 | 0 | 12 |
| 0 | 0 | 59 | 43 | 85 | 122 |
| 0 | 0 | 96 | 44 | 114 | 80 |
| 0 | 0 | 38 | 63 | 22 | 10 |
| 3 | 2 | 0 | 0 | 0 | 0 |
| 132 | 134 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 93 | 81 | 71 |
| 0 | 0 | 134 | 28 | 111 | 56 |
| 0 | 0 | 63 | 86 | 0 | 136 |
| 0 | 0 | 111 | 116 | 105 | 1 |
| 134 | 135 | 0 | 0 | 0 | 0 |
| 4 | 3 | 0 | 0 | 0 | 0 |
| 0 | 0 | 108 | 93 | 81 | 71 |
| 0 | 0 | 134 | 28 | 111 | 56 |
| 0 | 0 | 63 | 86 | 0 | 136 |
| 0 | 0 | 111 | 116 | 105 | 1 |
G:=sub<GL(6,GF(137))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,103,56,111,117,0,0,1,0,0,57,0,0,0,1,0,106,0,0,0,0,1,86],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,107,59,96,38,0,0,121,43,44,63,0,0,0,85,114,22,0,0,12,122,80,10],[3,132,0,0,0,0,2,134,0,0,0,0,0,0,108,134,63,111,0,0,93,28,86,116,0,0,81,111,0,105,0,0,71,56,136,1],[134,4,0,0,0,0,135,3,0,0,0,0,0,0,108,134,63,111,0,0,93,28,86,116,0,0,81,111,0,105,0,0,71,56,136,1] >;
D17.D4 in GAP, Magma, Sage, TeX
D_{17}.D_4 % in TeX
G:=Group("D17.D4"); // GroupNames label
G:=SmallGroup(272,35);
// by ID
G=gap.SmallGroup(272,35);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,101,5204,1614]);
// Polycyclic
G:=Group<a,b,c,d|a^17=b^2=c^4=1,d^2=a^-1*b,b*a*b=a^-1,c*a*c^-1=d*a*d^-1=a^4,c*b*c^-1=d*b*d^-1=a^3*b,d*c*d^-1=a^-1*b*c^-1>;
// generators/relations
Export
Subgroup lattice of D17.D4 in TeX
Character table of D17.D4 in TeX