direct product, abelian, monomial, 2-elementary
Aliases: C2xC34, SmallGroup(68,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C1 — C2xC34 |
| C1 — C2xC34 |
| C1 — C2xC34 |
Generators and relations for C2xC34
G = < a,b | a2=b34=1, ab=ba >
Quotients: C1, C2, C22, C17, C34, C2xC34
Smallest permutation representation of C2xC34
►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)]])
C2xC34 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 | C2xC34 | C34 | C22 | C2 |
| # reps | 1 | 3 | 16 | 48 |
Matrix representation of C2xC34 ►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] >;
C2xC34 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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