direct product, metacyclic, nilpotent (class 2), monomial, 2-elementary
Aliases: D4×C13, C4⋊C26, C52⋊3C2, C22⋊C26, C26.6C22, (C2×C26)⋊1C2, C2.1(C2×C26), SmallGroup(104,10)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×C13
G = < a,b,c | a13=b4=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of D4×C13
►On 52 points
Generators in S52
(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)
(1 44 25 32)(2 45 26 33)(3 46 14 34)(4 47 15 35)(5 48 16 36)(6 49 17 37)(7 50 18 38)(8 51 19 39)(9 52 20 27)(10 40 21 28)(11 41 22 29)(12 42 23 30)(13 43 24 31)
(27 52)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 48)(37 49)(38 50)(39 51)
G:=sub<Sym(52)| (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), (1,44,25,32)(2,45,26,33)(3,46,14,34)(4,47,15,35)(5,48,16,36)(6,49,17,37)(7,50,18,38)(8,51,19,39)(9,52,20,27)(10,40,21,28)(11,41,22,29)(12,42,23,30)(13,43,24,31), (27,52)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51)>;
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), (1,44,25,32)(2,45,26,33)(3,46,14,34)(4,47,15,35)(5,48,16,36)(6,49,17,37)(7,50,18,38)(8,51,19,39)(9,52,20,27)(10,40,21,28)(11,41,22,29)(12,42,23,30)(13,43,24,31), (27,52)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48)(37,49)(38,50)(39,51) );
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)], [(1,44,25,32),(2,45,26,33),(3,46,14,34),(4,47,15,35),(5,48,16,36),(6,49,17,37),(7,50,18,38),(8,51,19,39),(9,52,20,27),(10,40,21,28),(11,41,22,29),(12,42,23,30),(13,43,24,31)], [(27,52),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48),(37,49),(38,50),(39,51)]])
D4×C13 is a maximal subgroup of
D4⋊D13 D4.D13 D4⋊2D13
65 conjugacy classes
| class | 1 | 2A | 2B | 2C | 4 | 13A | ··· | 13L | 26A | ··· | 26L | 26M | ··· | 26AJ | 52A | ··· | 52L |
| order | 1 | 2 | 2 | 2 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | ··· | 1 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 |
65 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | ||||
| image | C1 | C2 | C2 | C13 | C26 | C26 | D4 | D4×C13 |
| kernel | D4×C13 | C52 | C2×C26 | D4 | C4 | C22 | C13 | C1 |
| # reps | 1 | 1 | 2 | 12 | 12 | 24 | 1 | 12 |
Matrix representation of D4×C13 ►in GL2(𝔽53) generated by
| 15 | 0 |
| 0 | 15 |
| 0 | 52 |
| 1 | 0 |
| 1 | 0 |
| 0 | 52 |
G:=sub<GL(2,GF(53))| [15,0,0,15],[0,1,52,0],[1,0,0,52] >;
D4×C13 in GAP, Magma, Sage, TeX
D_4\times C_{13} % in TeX
G:=Group("D4xC13"); // GroupNames label
G:=SmallGroup(104,10);
// by ID
G=gap.SmallGroup(104,10);
# by ID
G:=PCGroup([4,-2,-2,-13,-2,433]);
// Polycyclic
G:=Group<a,b,c|a^13=b^4=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
Export