direct product, abelian, monomial, 2-elementary
Aliases: C2×C34, SmallGroup(68,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2×C34 |
| C1 — C2×C34 |
| C1 — C2×C34 |
Generators and relations for C2×C34
G = < a,b | a2=b34=1, ab=ba >
Smallest permutation representation of C2×C34
►Regular action on 68 points
Generators in S68
(1 66)(2 67)(3 68)(4 35)(5 36)(6 37)(7 38)(8 39)(9 40)(10 41)(11 42)(12 43)(13 44)(14 45)(15 46)(16 47)(17 48)(18 49)(19 50)(20 51)(21 52)(22 53)(23 54)(24 55)(25 56)(26 57)(27 58)(28 59)(29 60)(30 61)(31 62)(32 63)(33 64)(34 65)
(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)
G:=sub<Sym(68)| (1,66)(2,67)(3,68)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65), (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)>;
G:=Group( (1,66)(2,67)(3,68)(4,35)(5,36)(6,37)(7,38)(8,39)(9,40)(10,41)(11,42)(12,43)(13,44)(14,45)(15,46)(16,47)(17,48)(18,49)(19,50)(20,51)(21,52)(22,53)(23,54)(24,55)(25,56)(26,57)(27,58)(28,59)(29,60)(30,61)(31,62)(32,63)(33,64)(34,65), (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) );
G=PermutationGroup([[(1,66),(2,67),(3,68),(4,35),(5,36),(6,37),(7,38),(8,39),(9,40),(10,41),(11,42),(12,43),(13,44),(14,45),(15,46),(16,47),(17,48),(18,49),(19,50),(20,51),(21,52),(22,53),(23,54),(24,55),(25,56),(26,57),(27,58),(28,59),(29,60),(30,61),(31,62),(32,63),(33,64),(34,65)], [(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)]])
C2×C34 is a maximal subgroup of
C17⋊D4
68 conjugacy classes
| class | 1 | 2A | 2B | 2C | 17A | ··· | 17P | 34A | ··· | 34AV |
| order | 1 | 2 | 2 | 2 | 17 | ··· | 17 | 34 | ··· | 34 |
| size | 1 | 1 | 1 | 1 | 1 | ··· | 1 | 1 | ··· | 1 |
68 irreducible representations
| dim | 1 | 1 | 1 | 1 |
| type | + | + | ||
| image | C1 | C2 | C17 | C34 |
| kernel | C2×C34 | C34 | C22 | C2 |
| # reps | 1 | 3 | 16 | 48 |
Matrix representation of C2×C34 ►in GL2(𝔽103) generated by
| 1 | 0 |
| 0 | 102 |
| 73 | 0 |
| 0 | 13 |
G:=sub<GL(2,GF(103))| [1,0,0,102],[73,0,0,13] >;
C2×C34 in GAP, Magma, Sage, TeX
C_2\times C_{34} % in TeX
G:=Group("C2xC34"); // GroupNames label
G:=SmallGroup(68,5);
// by ID
G=gap.SmallGroup(68,5);
# by ID
G:=PCGroup([3,-2,-2,-17]);
// Polycyclic
G:=Group<a,b|a^2=b^34=1,a*b=b*a>;
// generators/relations
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