direct product, metacyclic, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: C2×C17⋊C8, C34⋊C8, D17⋊C8, D34.C4, C17⋊(C2×C8), C17⋊C4.2C4, D17.(C2×C4), C17⋊C4.C22, (C2×C17⋊C4).2C2, SmallGroup(272,51)
Series: Derived ►Chief ►Lower central ►Upper central
| C17 — C2×C17⋊C8 |
Generators and relations for C2×C17⋊C8
G = < a,b,c | a2=b17=c8=1, ab=ba, ac=ca, cbc-1=b2 >
Character table of C2×C17⋊C8
| class | 1 | 2A | 2B | 2C | 4A | 4B | 4C | 4D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 17A | 17B | 34A | 34B | |
| size | 1 | 1 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 17 | 8 | 8 | 8 | 8 | |
| ρ1 | 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 | 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | i | i | i | -i | -i | -i | -i | i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | i | -i | -i | i | i | -i | -i | i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ7 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | -i | -i | i | i | i | i | -i | 1 | 1 | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -i | i | i | -i | -i | i | i | -i | 1 | 1 | 1 | 1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | i | -i | i | -i | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ10 | 1 | -1 | -1 | 1 | -i | -i | i | i | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ11 | 1 | -1 | -1 | 1 | i | i | -i | -i | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ12 | 1 | 1 | -1 | -1 | -i | i | -i | i | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ13 | 1 | -1 | -1 | 1 | i | i | -i | -i | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ14 | 1 | 1 | -1 | -1 | i | -i | i | -i | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | -1 | -i | i | -i | i | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | 1 | 1 | 1 | 1 | linear of order 8 |
| ρ16 | 1 | -1 | -1 | 1 | -i | -i | i | i | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | 1 | 1 | -1 | -1 | linear of order 8 |
| ρ17 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | 1+√17/2 | 1-√17/2 | orthogonal faithful |
| ρ18 | 8 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | 1-√17/2 | 1+√17/2 | orthogonal faithful |
| ρ19 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1-√17/2 | -1+√17/2 | -1+√17/2 | -1-√17/2 | orthogonal lifted from C17⋊C8 |
| ρ20 | 8 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1+√17/2 | -1-√17/2 | -1-√17/2 | -1+√17/2 | orthogonal lifted from C17⋊C8 |
►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 10 14 16 17 9 5 3)(4 11 6 12 15 8 13 7)(19 27 31 33 34 26 22 20)(21 28 23 29 32 25 30 24)
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), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24)>;
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), (2,10,14,16,17,9,5,3)(4,11,6,12,15,8,13,7)(19,27,31,33,34,26,22,20)(21,28,23,29,32,25,30,24) );
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)], [(2,10,14,16,17,9,5,3),(4,11,6,12,15,8,13,7),(19,27,31,33,34,26,22,20),(21,28,23,29,32,25,30,24)]])
Matrix representation of C2×C17⋊C8 ►in GL8(𝔽137)
| 136 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 136 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 136 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 136 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 136 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 136 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 136 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 136 |
| 2 | 1 | 22 | 115 | 136 | 135 | 114 | 136 |
| 3 | 1 | 22 | 115 | 136 | 135 | 114 | 136 |
| 2 | 2 | 22 | 115 | 136 | 135 | 114 | 136 |
| 2 | 1 | 23 | 115 | 136 | 135 | 114 | 136 |
| 2 | 1 | 22 | 116 | 136 | 135 | 114 | 136 |
| 2 | 1 | 22 | 115 | 0 | 135 | 114 | 136 |
| 2 | 1 | 22 | 115 | 136 | 136 | 114 | 136 |
| 2 | 1 | 22 | 115 | 136 | 135 | 115 | 136 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 20 | 114 | 116 | 43 | 117 | 1 | 45 | 116 |
| 67 | 2 | 95 | 64 | 94 | 114 | 41 | 92 |
| 43 | 1 | 94 | 65 | 93 | 115 | 42 | 116 |
| 135 | 136 | 115 | 22 | 1 | 2 | 23 | 1 |
G:=sub<GL(8,GF(137))| [136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136,0,0,0,0,0,0,0,0,136],[2,3,2,2,2,2,2,2,1,1,2,1,1,1,1,1,22,22,22,23,22,22,22,22,115,115,115,115,116,115,115,115,136,136,136,136,136,0,136,136,135,135,135,135,135,135,136,135,114,114,114,114,114,114,114,115,136,136,136,136,136,136,136,136],[0,0,0,0,20,67,43,135,1,0,0,0,114,2,1,136,0,0,0,0,116,95,94,115,0,1,0,0,43,64,65,22,0,0,0,0,117,94,93,1,0,0,1,0,1,114,115,2,0,0,0,0,45,41,42,23,0,0,0,1,116,92,116,1] >;
C2×C17⋊C8 in GAP, Magma, Sage, TeX
C_2\times C_{17}\rtimes C_8 % in TeX
G:=Group("C2xC17:C8"); // GroupNames label
G:=SmallGroup(272,51);
// by ID
G=gap.SmallGroup(272,51);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-17,20,42,3604,1314,819]);
// Polycyclic
G:=Group<a,b,c|a^2=b^17=c^8=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^2>;
// generators/relations
Export
Subgroup lattice of C2×C17⋊C8 in TeX
Character table of C2×C17⋊C8 in TeX