Copied to
clipboard

## G = C5×D13order 130 = 2·5·13

### Direct product of C5 and D13

Aliases: C5×D13, C13⋊C10, C652C2, SmallGroup(130,2)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C13 — C5×D13
 Chief series C1 — C13 — C65 — C5×D13
 Lower central C13 — C5×D13
 Upper central C1 — C5

Generators and relations for C5×D13
G = < a,b,c | a5=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

Smallest permutation representation of C5×D13
On 65 points
Generators in S65
(1 61 40 37 23)(2 62 41 38 24)(3 63 42 39 25)(4 64 43 27 26)(5 65 44 28 14)(6 53 45 29 15)(7 54 46 30 16)(8 55 47 31 17)(9 56 48 32 18)(10 57 49 33 19)(11 58 50 34 20)(12 59 51 35 21)(13 60 52 36 22)
(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 13)(2 12)(3 11)(4 10)(5 9)(6 8)(14 18)(15 17)(19 26)(20 25)(21 24)(22 23)(27 33)(28 32)(29 31)(34 39)(35 38)(36 37)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 55)(56 65)(57 64)(58 63)(59 62)(60 61)

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

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

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

C5×D13 is a maximal subgroup of   C65⋊C4

40 conjugacy classes

 class 1 2 5A 5B 5C 5D 10A 10B 10C 10D 13A ··· 13F 65A ··· 65X order 1 2 5 5 5 5 10 10 10 10 13 ··· 13 65 ··· 65 size 1 13 1 1 1 1 13 13 13 13 2 ··· 2 2 ··· 2

40 irreducible representations

 dim 1 1 1 1 2 2 type + + + image C1 C2 C5 C10 D13 C5×D13 kernel C5×D13 C65 D13 C13 C5 C1 # reps 1 1 4 4 6 24

Matrix representation of C5×D13 in GL2(𝔽131) generated by

 89 0 0 89
,
 0 1 130 123
,
 0 1 1 0
G:=sub<GL(2,GF(131))| [89,0,0,89],[0,130,1,123],[0,1,1,0] >;

C5×D13 in GAP, Magma, Sage, TeX

C_5\times D_{13}
% in TeX

G:=Group("C5xD13");
// GroupNames label

G:=SmallGroup(130,2);
// by ID

G=gap.SmallGroup(130,2);
# by ID

G:=PCGroup([3,-2,-5,-13,1082]);
// Polycyclic

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

Export

׿
×
𝔽