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## G = C13×F5order 260 = 22·5·13

### Direct product of C13 and F5

Aliases: C13×F5, C5⋊C52, C656C4, D5.C26, (D5×C13).2C2, SmallGroup(260,7)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — C13×F5
 Chief series C1 — C5 — D5 — D5×C13 — C13×F5
 Lower central C5 — C13×F5
 Upper central C1 — C13

Generators and relations for C13×F5
G = < a,b,c | a13=b5=c4=1, ab=ba, ac=ca, cbc-1=b3 >

Smallest permutation representation of C13×F5
On 65 points
Generators in S65
(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)
(1 37 65 24 52)(2 38 53 25 40)(3 39 54 26 41)(4 27 55 14 42)(5 28 56 15 43)(6 29 57 16 44)(7 30 58 17 45)(8 31 59 18 46)(9 32 60 19 47)(10 33 61 20 48)(11 34 62 21 49)(12 35 63 22 50)(13 36 64 23 51)
(14 27 55 42)(15 28 56 43)(16 29 57 44)(17 30 58 45)(18 31 59 46)(19 32 60 47)(20 33 61 48)(21 34 62 49)(22 35 63 50)(23 36 64 51)(24 37 65 52)(25 38 53 40)(26 39 54 41)

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

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

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

65 conjugacy classes

 class 1 2 4A 4B 5 13A ··· 13L 26A ··· 26L 52A ··· 52X 65A ··· 65L order 1 2 4 4 5 13 ··· 13 26 ··· 26 52 ··· 52 65 ··· 65 size 1 5 5 5 4 1 ··· 1 5 ··· 5 5 ··· 5 4 ··· 4

65 irreducible representations

 dim 1 1 1 1 1 1 4 4 type + + + image C1 C2 C4 C13 C26 C52 F5 C13×F5 kernel C13×F5 D5×C13 C65 F5 D5 C5 C13 C1 # reps 1 1 2 12 12 24 1 12

Matrix representation of C13×F5 in GL4(𝔽521) generated by

 302 0 0 0 0 302 0 0 0 0 302 0 0 0 0 302
,
 0 0 0 520 1 0 0 520 0 1 0 520 0 0 1 520
,
 0 0 1 0 1 0 0 0 0 0 0 1 0 1 0 0
G:=sub<GL(4,GF(521))| [302,0,0,0,0,302,0,0,0,0,302,0,0,0,0,302],[0,1,0,0,0,0,1,0,0,0,0,1,520,520,520,520],[0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0] >;

C13×F5 in GAP, Magma, Sage, TeX

C_{13}\times F_5
% in TeX

G:=Group("C13xF5");
// GroupNames label

G:=SmallGroup(260,7);
// by ID

G=gap.SmallGroup(260,7);
# by ID

G:=PCGroup([4,-2,-13,-2,-5,104,1667,139]);
// Polycyclic

G:=Group<a,b,c|a^13=b^5=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^3>;
// generators/relations

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