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G = C5xD13order 130 = 2·5·13

Direct product of C5 and D13

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

Aliases: C5xD13, C13:C10, C65:2C2, SmallGroup(130,2)

Series: Derived Chief Lower central Upper central

C1C13 — C5xD13
C1C13C65 — C5xD13
C13 — C5xD13
C1C5

Generators and relations for C5xD13
 G = < a,b,c | a5=b13=c2=1, ab=ba, ac=ca, cbc=b-1 >

Subgroups: 32 in 8 conjugacy classes, 6 normal (all characteristic)
Quotients: C1, C2, C5, C10, D13, C5xD13
13C2
13C10

Smallest permutation representation of C5xD13
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)]])

C5xD13 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+++
imageC1C2C5C10D13C5xD13
kernelC5xD13C65D13C13C5C1
# reps1144624

Matrix representation of C5xD13 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] >;

C5xD13 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 C5xD13 in TeX

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