Copied to
clipboard

G = C5×D13order 130 = 2·5·13

Direct product of C5 and D13

direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary

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

Series: Derived Chief Lower central Upper central

C1C13 — C5×D13
C1C13C65 — C5×D13
C13 — C5×D13
C1C5

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

13C2
13C10

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

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

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

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

C5×D13 is a maximal subgroup of   C65⋊C4

40 conjugacy classes

class 1  2 5A5B5C5D10A10B10C10D13A···13F65A···65X
order1255551010101013···1365···65
size1131111131313132···22···2

40 irreducible representations

dim111122
type+++
imageC1C2C5C10D13C5×D13
kernelC5×D13C65D13C13C5C1
# reps1144624

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

890
089
,
01
130123
,
01
10
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

Subgroup lattice of C5×D13 in TeX

׿
×
𝔽