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## G = C2×C26order 52 = 22·13

### Abelian group of type [2,26]

Aliases: C2×C26, SmallGroup(52,5)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C26
 Chief series C1 — C13 — C26 — C2×C26
 Lower central C1 — C2×C26
 Upper central C1 — C2×C26

Generators and relations for C2×C26
G = < a,b | a2=b26=1, ab=ba >

Smallest permutation representation of C2×C26
Regular action on 52 points
Generators in S52
(1 27)(2 28)(3 29)(4 30)(5 31)(6 32)(7 33)(8 34)(9 35)(10 36)(11 37)(12 38)(13 39)(14 40)(15 41)(16 42)(17 43)(18 44)(19 45)(20 46)(21 47)(22 48)(23 49)(24 50)(25 51)(26 52)
(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)

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

G:=Group( (1,27)(2,28)(3,29)(4,30)(5,31)(6,32)(7,33)(8,34)(9,35)(10,36)(11,37)(12,38)(13,39)(14,40)(15,41)(16,42)(17,43)(18,44)(19,45)(20,46)(21,47)(22,48)(23,49)(24,50)(25,51)(26,52), (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) );

G=PermutationGroup([(1,27),(2,28),(3,29),(4,30),(5,31),(6,32),(7,33),(8,34),(9,35),(10,36),(11,37),(12,38),(13,39),(14,40),(15,41),(16,42),(17,43),(18,44),(19,45),(20,46),(21,47),(22,48),(23,49),(24,50),(25,51),(26,52)], [(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)])

52 conjugacy classes

 class 1 2A 2B 2C 13A ··· 13L 26A ··· 26AJ order 1 2 2 2 13 ··· 13 26 ··· 26 size 1 1 1 1 1 ··· 1 1 ··· 1

52 irreducible representations

 dim 1 1 1 1 type + + image C1 C2 C13 C26 kernel C2×C26 C26 C22 C2 # reps 1 3 12 36

Matrix representation of C2×C26 in GL2(𝔽53) generated by

 52 0 0 52
,
 10 0 0 38
G:=sub<GL(2,GF(53))| [52,0,0,52],[10,0,0,38] >;

C2×C26 in GAP, Magma, Sage, TeX

C_2\times C_{26}
% in TeX

G:=Group("C2xC26");
// GroupNames label

G:=SmallGroup(52,5);
// by ID

G=gap.SmallGroup(52,5);
# by ID

G:=PCGroup([3,-2,-2,-13]);
// Polycyclic

G:=Group<a,b|a^2=b^26=1,a*b=b*a>;
// generators/relations

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