direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: D5×C13, C5⋊C26, C65⋊3C2, SmallGroup(130,1)
Series: Derived ►Chief ►Lower central ►Upper central
| C5 — D5×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 47 64 30 25)(2 48 65 31 26)(3 49 53 32 14)(4 50 54 33 15)(5 51 55 34 16)(6 52 56 35 17)(7 40 57 36 18)(8 41 58 37 19)(9 42 59 38 20)(10 43 60 39 21)(11 44 61 27 22)(12 45 62 28 23)(13 46 63 29 24)
(1 25)(2 26)(3 14)(4 15)(5 16)(6 17)(7 18)(8 19)(9 20)(10 21)(11 22)(12 23)(13 24)(27 44)(28 45)(29 46)(30 47)(31 48)(32 49)(33 50)(34 51)(35 52)(36 40)(37 41)(38 42)(39 43)
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), (1,47,64,30,25)(2,48,65,31,26)(3,49,53,32,14)(4,50,54,33,15)(5,51,55,34,16)(6,52,56,35,17)(7,40,57,36,18)(8,41,58,37,19)(9,42,59,38,20)(10,43,60,39,21)(11,44,61,27,22)(12,45,62,28,23)(13,46,63,29,24), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43)>;
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), (1,47,64,30,25)(2,48,65,31,26)(3,49,53,32,14)(4,50,54,33,15)(5,51,55,34,16)(6,52,56,35,17)(7,40,57,36,18)(8,41,58,37,19)(9,42,59,38,20)(10,43,60,39,21)(11,44,61,27,22)(12,45,62,28,23)(13,46,63,29,24), (1,25)(2,26)(3,14)(4,15)(5,16)(6,17)(7,18)(8,19)(9,20)(10,21)(11,22)(12,23)(13,24)(27,44)(28,45)(29,46)(30,47)(31,48)(32,49)(33,50)(34,51)(35,52)(36,40)(37,41)(38,42)(39,43) );
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)], [(1,47,64,30,25),(2,48,65,31,26),(3,49,53,32,14),(4,50,54,33,15),(5,51,55,34,16),(6,52,56,35,17),(7,40,57,36,18),(8,41,58,37,19),(9,42,59,38,20),(10,43,60,39,21),(11,44,61,27,22),(12,45,62,28,23),(13,46,63,29,24)], [(1,25),(2,26),(3,14),(4,15),(5,16),(6,17),(7,18),(8,19),(9,20),(10,21),(11,22),(12,23),(13,24),(27,44),(28,45),(29,46),(30,47),(31,48),(32,49),(33,50),(34,51),(35,52),(36,40),(37,41),(38,42),(39,43)]])
D5×C13 is a maximal subgroup of
C13⋊3F5
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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