direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C17, C4⋊C34, C68⋊3C2, C22⋊C34, C34.6C22, (C2×C34)⋊1C2, C2.1(C2×C34), SmallGroup(136,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C17
G = < a,b,c | a17=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C17
►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 62 36 24)(2 63 37 25)(3 64 38 26)(4 65 39 27)(5 66 40 28)(6 67 41 29)(7 68 42 30)(8 52 43 31)(9 53 44 32)(10 54 45 33)(11 55 46 34)(12 56 47 18)(13 57 48 19)(14 58 49 20)(15 59 50 21)(16 60 51 22)(17 61 35 23)
(18 56)(19 57)(20 58)(21 59)(22 60)(23 61)(24 62)(25 63)(26 64)(27 65)(28 66)(29 67)(30 68)(31 52)(32 53)(33 54)(34 55)
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), (1,62,36,24)(2,63,37,25)(3,64,38,26)(4,65,39,27)(5,66,40,28)(6,67,41,29)(7,68,42,30)(8,52,43,31)(9,53,44,32)(10,54,45,33)(11,55,46,34)(12,56,47,18)(13,57,48,19)(14,58,49,20)(15,59,50,21)(16,60,51,22)(17,61,35,23), (18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,52)(32,53)(33,54)(34,55)>;
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), (1,62,36,24)(2,63,37,25)(3,64,38,26)(4,65,39,27)(5,66,40,28)(6,67,41,29)(7,68,42,30)(8,52,43,31)(9,53,44,32)(10,54,45,33)(11,55,46,34)(12,56,47,18)(13,57,48,19)(14,58,49,20)(15,59,50,21)(16,60,51,22)(17,61,35,23), (18,56)(19,57)(20,58)(21,59)(22,60)(23,61)(24,62)(25,63)(26,64)(27,65)(28,66)(29,67)(30,68)(31,52)(32,53)(33,54)(34,55) );
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)], [(1,62,36,24),(2,63,37,25),(3,64,38,26),(4,65,39,27),(5,66,40,28),(6,67,41,29),(7,68,42,30),(8,52,43,31),(9,53,44,32),(10,54,45,33),(11,55,46,34),(12,56,47,18),(13,57,48,19),(14,58,49,20),(15,59,50,21),(16,60,51,22),(17,61,35,23)], [(18,56),(19,57),(20,58),(21,59),(22,60),(23,61),(24,62),(25,63),(26,64),(27,65),(28,66),(29,67),(30,68),(31,52),(32,53),(33,54),(34,55)]])
D4×C17 is a maximal subgroup of
D4⋊D17 D4.D17 D4⋊2D17
85 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 17A | ··· | 17P | 34A | ··· | 34P | 34Q | ··· | 34AV | 68A | ··· | 68P |
| order | 1 | 2 | 2 | 2 | 4 | 17 | ··· | 17 | 34 | ··· | 34 | 34 | ··· | 34 | 68 | ··· | 68 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
85 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C17 | C34 | C34 | D4 | D4×C17 |
| kernel | D4×C17 | C68 | C2×C34 | D4 | C4 | C22 | C17 | C1 |
| # reps | 1 | 1 | 2 | 16 | 16 | 32 | 1 | 16 |
Matrix representation of D4×C17 ►in GL2(𝔽137) generated by
| 50 | 0 |
| 0 | 50 |
| 0 | 136 |
| 1 | 0 |
| 1 | 0 |
| 0 | 136 |
G:=sub<GL(2,GF(137))| [50,0,0,50],[0,1,136,0],[1,0,0,136] >;
D4×C17 in GAP, Magma, Sage, TeX
D_4\times C_{17} % in TeX
G:=Group("D4xC17"); // GroupNames label
G:=SmallGroup(136,10);
// by ID
G=gap.SmallGroup(136,10);
# by ID
G:=PCGroup([4,-2,-2,-17,-2,561]);
// Polycyclic
G:=Group<a,b,c|a^17=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export