direct product, metabelian, supersoluble, monomial, A-group, 2-hyperelementary
Aliases: D5×D13, D65⋊C2, C5⋊1D26, C65⋊C22, C13⋊1D10, (D5×C13)⋊C2, (C5×D13)⋊C2, SmallGroup(260,11)
Series: Derived ►Chief ►Lower central ►Upper central
| C65 — D5×D13 |
Generators and relations for D5×D13
G = < a,b,c,d | a5=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Smallest permutation representation of D5×D13
►On 65 points
Generators in S65
(1 56 40 33 23)(2 57 41 34 24)(3 58 42 35 25)(4 59 43 36 26)(5 60 44 37 14)(6 61 45 38 15)(7 62 46 39 16)(8 63 47 27 17)(9 64 48 28 18)(10 65 49 29 19)(11 53 50 30 20)(12 54 51 31 21)(13 55 52 32 22)
(1 23)(2 24)(3 25)(4 26)(5 14)(6 15)(7 16)(8 17)(9 18)(10 19)(11 20)(12 21)(13 22)(27 63)(28 64)(29 65)(30 53)(31 54)(32 55)(33 56)(34 57)(35 58)(36 59)(37 60)(38 61)(39 62)
(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 38)(28 37)(29 36)(30 35)(31 34)(32 33)(40 52)(41 51)(42 50)(43 49)(44 48)(45 47)(53 58)(54 57)(55 56)(59 65)(60 64)(61 63)
G:=sub<Sym(65)| (1,56,40,33,23)(2,57,41,34,24)(3,58,42,35,25)(4,59,43,36,26)(5,60,44,37,14)(6,61,45,38,15)(7,62,46,39,16)(8,63,47,27,17)(9,64,48,28,18)(10,65,49,29,19)(11,53,50,30,20)(12,54,51,31,21)(13,55,52,32,22), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,63)(28,64)(29,65)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62), (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)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63)>;
G:=Group( (1,56,40,33,23)(2,57,41,34,24)(3,58,42,35,25)(4,59,43,36,26)(5,60,44,37,14)(6,61,45,38,15)(7,62,46,39,16)(8,63,47,27,17)(9,64,48,28,18)(10,65,49,29,19)(11,53,50,30,20)(12,54,51,31,21)(13,55,52,32,22), (1,23)(2,24)(3,25)(4,26)(5,14)(6,15)(7,16)(8,17)(9,18)(10,19)(11,20)(12,21)(13,22)(27,63)(28,64)(29,65)(30,53)(31,54)(32,55)(33,56)(34,57)(35,58)(36,59)(37,60)(38,61)(39,62), (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)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(40,52)(41,51)(42,50)(43,49)(44,48)(45,47)(53,58)(54,57)(55,56)(59,65)(60,64)(61,63) );
G=PermutationGroup([[(1,56,40,33,23),(2,57,41,34,24),(3,58,42,35,25),(4,59,43,36,26),(5,60,44,37,14),(6,61,45,38,15),(7,62,46,39,16),(8,63,47,27,17),(9,64,48,28,18),(10,65,49,29,19),(11,53,50,30,20),(12,54,51,31,21),(13,55,52,32,22)], [(1,23),(2,24),(3,25),(4,26),(5,14),(6,15),(7,16),(8,17),(9,18),(10,19),(11,20),(12,21),(13,22),(27,63),(28,64),(29,65),(30,53),(31,54),(32,55),(33,56),(34,57),(35,58),(36,59),(37,60),(38,61),(39,62)], [(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),(27,38),(28,37),(29,36),(30,35),(31,34),(32,33),(40,52),(41,51),(42,50),(43,49),(44,48),(45,47),(53,58),(54,57),(55,56),(59,65),(60,64),(61,63)]])
32 conjugacy classes
| class | 1 | 2A | 2B | 2C | 5A | 5B | 10A | 10B | 13A | ··· | 13F | 26A | ··· | 26F | 65A | ··· | 65L |
| order | 1 | 2 | 2 | 2 | 5 | 5 | 10 | 10 | 13 | ··· | 13 | 26 | ··· | 26 | 65 | ··· | 65 |
| size | 1 | 5 | 13 | 65 | 2 | 2 | 26 | 26 | 2 | ··· | 2 | 10 | ··· | 10 | 4 | ··· | 4 |
32 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | D5 | D10 | D13 | D26 | D5×D13 |
| kernel | D5×D13 | D5×C13 | C5×D13 | D65 | D13 | C13 | D5 | C5 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 12 |
Matrix representation of D5×D13 ►in GL4(𝔽131) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 130 | 119 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
| 103 | 1 | 0 | 0 |
| 30 | 41 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 106 | 75 | 0 | 0 |
| 86 | 25 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(131))| [1,0,0,0,0,1,0,0,0,0,0,130,0,0,1,119],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,1,0],[103,30,0,0,1,41,0,0,0,0,1,0,0,0,0,1],[106,86,0,0,75,25,0,0,0,0,1,0,0,0,0,1] >;
D5×D13 in GAP, Magma, Sage, TeX
D_5\times D_{13} % in TeX
G:=Group("D5xD13"); // GroupNames label
G:=SmallGroup(260,11);
// by ID
G=gap.SmallGroup(260,11);
# by ID
G:=PCGroup([4,-2,-2,-5,-13,102,3843]);
// Polycyclic
G:=Group<a,b,c,d|a^5=b^2=c^13=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations
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