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 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