direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×D13, C13⋊C10, C65⋊2C2, SmallGroup(130,2)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C5×D13 |
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 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 | 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