direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C17⋊C4, C34⋊C4, D17⋊C4, D34.C2, D17.C22, C17⋊(C2×C4), SmallGroup(136,13)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×C17⋊C4 |
Generators and relations for C2×C17⋊C4
G = < a,b,c | a2=b17=c4=1, ab=ba, ac=ca, cbc-1=b4 >
Character table of C2×C17⋊C4
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 17A | 17B | 17C | 17D | 34A | 34B | 34C | 34D | |
| size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 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 | trivial |
| ρ2 | 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 | linear of order 2 |
| ρ4 | 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 | i | i | -i | -i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | -i | i | i | -i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -i | -i | i | i | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | i | -i | -i | i | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
| ρ9 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1715-ζ179-ζ178-ζ172 | orthogonal faithful |
| ρ10 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1714-ζ1712-ζ175-ζ173 | orthogonal faithful |
| ρ11 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | orthogonal lifted from C17⋊C4 |
| ρ12 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1714+ζ1712+ζ175+ζ173 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1711+ζ1710+ζ177+ζ176 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1716-ζ1713-ζ174-ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1711-ζ1710-ζ177-ζ176 | orthogonal faithful |
| ρ13 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1711+ζ1710+ζ177+ζ176 | ζ1714+ζ1712+ζ175+ζ173 | ζ1715+ζ179+ζ178+ζ172 | orthogonal lifted from C17⋊C4 |
| ρ14 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1715+ζ179+ζ178+ζ172 | ζ1716+ζ1713+ζ174+ζ17 | ζ1714+ζ1712+ζ175+ζ173 | orthogonal lifted from C17⋊C4 |
| ρ15 | 4 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | orthogonal lifted from C17⋊C4 |
| ρ16 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | ζ1715+ζ179+ζ178+ζ172 | ζ1714+ζ1712+ζ175+ζ173 | ζ1711+ζ1710+ζ177+ζ176 | ζ1716+ζ1713+ζ174+ζ17 | -ζ1715-ζ179-ζ178-ζ172 | -ζ1714-ζ1712-ζ175-ζ173 | -ζ1711-ζ1710-ζ177-ζ176 | -ζ1716-ζ1713-ζ174-ζ17 | orthogonal faithful |
►On 34 points
Generators in S34
(1 18)(2 19)(3 20)(4 21)(5 22)(6 23)(7 24)(8 25)(9 26)(10 27)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)
(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)
(2 14 17 5)(3 10 16 9)(4 6 15 13)(7 11 12 8)(19 31 34 22)(20 27 33 26)(21 23 32 30)(24 28 29 25)
G:=sub<Sym(34)| (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25)>;
G:=Group( (1,18)(2,19)(3,20)(4,21)(5,22)(6,23)(7,24)(8,25)(9,26)(10,27)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34), (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), (2,14,17,5)(3,10,16,9)(4,6,15,13)(7,11,12,8)(19,31,34,22)(20,27,33,26)(21,23,32,30)(24,28,29,25) );
G=PermutationGroup([[(1,18),(2,19),(3,20),(4,21),(5,22),(6,23),(7,24),(8,25),(9,26),(10,27),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34)], [(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)], [(2,14,17,5),(3,10,16,9),(4,6,15,13),(7,11,12,8),(19,31,34,22),(20,27,33,26),(21,23,32,30),(24,28,29,25)]])
C2×C17⋊C4 is a maximal subgroup of
C68⋊C4 D17.D4
C2×C17⋊C4 is a maximal quotient of C68.C4 D34.4C4 C68⋊C4 C17⋊M4(2) D17.D4
Matrix representation of C2×C17⋊C4 ►in GL4(𝔽137) generated by
| 136 | 0 | 0 | 0 |
| 0 | 136 | 0 | 0 |
| 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 136 |
| 75 | 114 | 74 | 136 |
| 7 | 42 | 121 | 115 |
| 32 | 88 | 6 | 111 |
| 119 | 11 | 49 | 66 |
| 114 | 77 | 107 | 132 |
| 54 | 49 | 15 | 18 |
| 20 | 112 | 93 | 72 |
| 105 | 11 | 67 | 18 |
G:=sub<GL(4,GF(137))| [136,0,0,0,0,136,0,0,0,0,136,0,0,0,0,136],[75,7,32,119,114,42,88,11,74,121,6,49,136,115,111,66],[114,54,20,105,77,49,112,11,107,15,93,67,132,18,72,18] >;
C2×C17⋊C4 in GAP, Magma, Sage, TeX
C_2\times C_{17}\rtimes C_4 % in TeX
G:=Group("C2xC17:C4"); // GroupNames label
G:=SmallGroup(136,13);
// by ID
G=gap.SmallGroup(136,13);
# by ID
G:=PCGroup([4,-2,-2,-2,-17,16,1667,523]);
// Polycyclic
G:=Group<a,b,c|a^2=b^17=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^4>;
// generators/relations
Export
Subgroup lattice of C2×C17⋊C4 in TeX
Character table of C2×C17⋊C4 in TeX