direct product, metacyclic, supersoluble, monomial, Z-group, 2-hyperelementary
Aliases: C5×C13⋊C4, C13⋊C20, C65⋊5C4, D13.C10, (C5×D13).2C2, SmallGroup(260,5)
Series: Derived ►Chief ►Lower central ►Upper central
| C13 — C5×C13⋊C4 |
Generators and relations for C5×C13⋊C4
G = < a,b,c | a5=b13=c4=1, ab=ba, ac=ca, cbc-1=b5 >
Smallest permutation representation of C5×C13⋊C4
►On 65 points
Generators in S65
(1 53 40 27 14)(2 54 41 28 15)(3 55 42 29 16)(4 56 43 30 17)(5 57 44 31 18)(6 58 45 32 19)(7 59 46 33 20)(8 60 47 34 21)(9 61 48 35 22)(10 62 49 36 23)(11 63 50 37 24)(12 64 51 38 25)(13 65 52 39 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)
(2 9 13 6)(3 4 12 11)(5 7 10 8)(15 22 26 19)(16 17 25 24)(18 20 23 21)(28 35 39 32)(29 30 38 37)(31 33 36 34)(41 48 52 45)(42 43 51 50)(44 46 49 47)(54 61 65 58)(55 56 64 63)(57 59 62 60)
G:=sub<Sym(65)| (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60)>;
G:=Group( (1,53,40,27,14)(2,54,41,28,15)(3,55,42,29,16)(4,56,43,30,17)(5,57,44,31,18)(6,58,45,32,19)(7,59,46,33,20)(8,60,47,34,21)(9,61,48,35,22)(10,62,49,36,23)(11,63,50,37,24)(12,64,51,38,25)(13,65,52,39,26), (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), (2,9,13,6)(3,4,12,11)(5,7,10,8)(15,22,26,19)(16,17,25,24)(18,20,23,21)(28,35,39,32)(29,30,38,37)(31,33,36,34)(41,48,52,45)(42,43,51,50)(44,46,49,47)(54,61,65,58)(55,56,64,63)(57,59,62,60) );
G=PermutationGroup([[(1,53,40,27,14),(2,54,41,28,15),(3,55,42,29,16),(4,56,43,30,17),(5,57,44,31,18),(6,58,45,32,19),(7,59,46,33,20),(8,60,47,34,21),(9,61,48,35,22),(10,62,49,36,23),(11,63,50,37,24),(12,64,51,38,25),(13,65,52,39,26)], [(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)], [(2,9,13,6),(3,4,12,11),(5,7,10,8),(15,22,26,19),(16,17,25,24),(18,20,23,21),(28,35,39,32),(29,30,38,37),(31,33,36,34),(41,48,52,45),(42,43,51,50),(44,46,49,47),(54,61,65,58),(55,56,64,63),(57,59,62,60)]])
35 conjugacy classes
| class | 1 | 2 | 4A | 4B | 5A | 5B | 5C | 5D | 10A | 10B | 10C | 10D | 13A | 13B | 13C | 20A | ··· | 20H | 65A | ··· | 65L |
| order | 1 | 2 | 4 | 4 | 5 | 5 | 5 | 5 | 10 | 10 | 10 | 10 | 13 | 13 | 13 | 20 | ··· | 20 | 65 | ··· | 65 |
| size | 1 | 13 | 13 | 13 | 1 | 1 | 1 | 1 | 13 | 13 | 13 | 13 | 4 | 4 | 4 | 13 | ··· | 13 | 4 | ··· | 4 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 |
| type | + | + | + | |||||
| image | C1 | C2 | C4 | C5 | C10 | C20 | C13⋊C4 | C5×C13⋊C4 |
| kernel | C5×C13⋊C4 | C5×D13 | C65 | C13⋊C4 | D13 | C13 | C5 | C1 |
| # reps | 1 | 1 | 2 | 4 | 4 | 8 | 3 | 12 |
Matrix representation of C5×C13⋊C4 ►in GL4(𝔽521) generated by
| 516 | 0 | 0 | 0 |
| 0 | 516 | 0 | 0 |
| 0 | 0 | 516 | 0 |
| 0 | 0 | 0 | 516 |
| 27 | 368 | 152 | 520 |
| 182 | 190 | 305 | 495 |
| 54 | 216 | 304 | 519 |
| 245 | 1 | 126 | 126 |
| 115 | 378 | 118 | 38 |
| 358 | 234 | 285 | 101 |
| 483 | 36 | 485 | 216 |
| 134 | 420 | 252 | 208 |
G:=sub<GL(4,GF(521))| [516,0,0,0,0,516,0,0,0,0,516,0,0,0,0,516],[27,182,54,245,368,190,216,1,152,305,304,126,520,495,519,126],[115,358,483,134,378,234,36,420,118,285,485,252,38,101,216,208] >;
C5×C13⋊C4 in GAP, Magma, Sage, TeX
C_5\times C_{13}\rtimes C_4 % in TeX
G:=Group("C5xC13:C4"); // GroupNames label
G:=SmallGroup(260,5);
// by ID
G=gap.SmallGroup(260,5);
# by ID
G:=PCGroup([4,-2,-5,-2,-13,40,2563,395]);
// Polycyclic
G:=Group<a,b,c|a^5=b^13=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^5>;
// generators/relations
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