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G = C2×C34order 68 = 22·17

Abelian group of type [2,34]

direct product, abelian, monomial, 2-elementary

Aliases: C2×C34, SmallGroup(68,5)

Series: Derived Chief Lower central Upper central

C1 — C2×C34
C1C17C34 — 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 35)(2 36)(3 37)(4 38)(5 39)(6 40)(7 41)(8 42)(9 43)(10 44)(11 45)(12 46)(13 47)(14 48)(15 49)(16 50)(17 51)(18 52)(19 53)(20 54)(21 55)(22 56)(23 57)(24 58)(25 59)(26 60)(27 61)(28 62)(29 63)(30 64)(31 65)(32 66)(33 67)(34 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)

G:=sub<Sym(68)| (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68), (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)>;

G:=Group( (1,35)(2,36)(3,37)(4,38)(5,39)(6,40)(7,41)(8,42)(9,43)(10,44)(11,45)(12,46)(13,47)(14,48)(15,49)(16,50)(17,51)(18,52)(19,53)(20,54)(21,55)(22,56)(23,57)(24,58)(25,59)(26,60)(27,61)(28,62)(29,63)(30,64)(31,65)(32,66)(33,67)(34,68), (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) );

G=PermutationGroup([(1,35),(2,36),(3,37),(4,38),(5,39),(6,40),(7,41),(8,42),(9,43),(10,44),(11,45),(12,46),(13,47),(14,48),(15,49),(16,50),(17,51),(18,52),(19,53),(20,54),(21,55),(22,56),(23,57),(24,58),(25,59),(26,60),(27,61),(28,62),(29,63),(30,64),(31,65),(32,66),(33,67),(34,68)], [(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)])

C2×C34 is a maximal subgroup of   C17⋊D4

68 conjugacy classes

class 1 2A2B2C17A···17P34A···34AV
order122217···1734···34
size11111···11···1

68 irreducible representations

dim1111
type++
imageC1C2C17C34
kernelC2×C34C34C22C2
# reps131648

Matrix representation of C2×C34 in GL2(𝔽103) generated by

10
0102
,
730
013
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

Export

Subgroup lattice of C2×C34 in TeX

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