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## G = D5×C13order 130 = 2·5·13

### Direct product of C13 and D5

Aliases: D5×C13, C5⋊C26, C653C2, SmallGroup(130,1)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5 — D5×C13
 Chief series C1 — C5 — C65 — D5×C13
 Lower central C5 — D5×C13
 Upper central C1 — C13

Generators and relations for D5×C13
G = < a,b,c | a13=b5=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of D5×C13
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 51 28 63 17)(2 52 29 64 18)(3 40 30 65 19)(4 41 31 53 20)(5 42 32 54 21)(6 43 33 55 22)(7 44 34 56 23)(8 45 35 57 24)(9 46 36 58 25)(10 47 37 59 26)(11 48 38 60 14)(12 49 39 61 15)(13 50 27 62 16)
(1 17)(2 18)(3 19)(4 20)(5 21)(6 22)(7 23)(8 24)(9 25)(10 26)(11 14)(12 15)(13 16)(40 65)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)(49 61)(50 62)(51 63)(52 64)

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

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

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

D5×C13 is a maximal subgroup of   C133F5

52 conjugacy classes

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

52 irreducible representations

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

Matrix representation of D5×C13 in GL2(𝔽131) generated by

 62 0 0 62
,
 130 1 10 120
,
 130 0 10 1
G:=sub<GL(2,GF(131))| [62,0,0,62],[130,10,1,120],[130,10,0,1] >;

D5×C13 in GAP, Magma, Sage, TeX

D_5\times C_{13}
% in TeX

G:=Group("D5xC13");
// GroupNames label

G:=SmallGroup(130,1);
// by ID

G=gap.SmallGroup(130,1);
# by ID

G:=PCGroup([3,-2,-13,-5,938]);
// Polycyclic

G:=Group<a,b,c|a^13=b^5=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations

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