direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4×D13, C4⋊1D26, C52⋊C22, D52⋊3C2, C22⋊1D26, D26⋊2C22, C26.5C23, Dic13⋊1C22, C13⋊2(C2×D4), (C2×C26)⋊C22, (C4×D13)⋊1C2, (D4×C13)⋊2C2, C13⋊D4⋊1C2, (C22×D13)⋊2C2, C2.6(C22×D13), SmallGroup(208,39)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4×D13
G = < a,b,c,d | a4=b2=c13=d2=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 370 in 54 conjugacy classes, 25 normal (13 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2×C4, D4, D4, C23, C13, C2×D4, D13, D13, C26, C26, Dic13, C52, D26, D26, D26, C2×C26, C4×D13, D52, C13⋊D4, D4×C13, C22×D13, D4×D13
Quotients: C1, C2, C22, D4, C23, C2×D4, D13, D26, C22×D13, D4×D13
►On 52 points
(1 37 19 43)(2 38 20 44)(3 39 21 45)(4 27 22 46)(5 28 23 47)(6 29 24 48)(7 30 25 49)(8 31 26 50)(9 32 14 51)(10 33 15 52)(11 34 16 40)(12 35 17 41)(13 36 18 42)
(27 46)(28 47)(29 48)(30 49)(31 50)(32 51)(33 52)(34 40)(35 41)(36 42)(37 43)(38 44)(39 45)
(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 18)(2 17)(3 16)(4 15)(5 14)(6 26)(7 25)(8 24)(9 23)(10 22)(11 21)(12 20)(13 19)(27 52)(28 51)(29 50)(30 49)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)
G:=sub<Sym(52)| (1,37,19,43)(2,38,20,44)(3,39,21,45)(4,27,22,46)(5,28,23,47)(6,29,24,48)(7,30,25,49)(8,31,26,50)(9,32,14,51)(10,33,15,52)(11,34,16,40)(12,35,17,41)(13,36,18,42), (27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)>;
G:=Group( (1,37,19,43)(2,38,20,44)(3,39,21,45)(4,27,22,46)(5,28,23,47)(6,29,24,48)(7,30,25,49)(8,31,26,50)(9,32,14,51)(10,33,15,52)(11,34,16,40)(12,35,17,41)(13,36,18,42), (27,46)(28,47)(29,48)(30,49)(31,50)(32,51)(33,52)(34,40)(35,41)(36,42)(37,43)(38,44)(39,45), (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), (1,18)(2,17)(3,16)(4,15)(5,14)(6,26)(7,25)(8,24)(9,23)(10,22)(11,21)(12,20)(13,19)(27,52)(28,51)(29,50)(30,49)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40) );
G=PermutationGroup([[(1,37,19,43),(2,38,20,44),(3,39,21,45),(4,27,22,46),(5,28,23,47),(6,29,24,48),(7,30,25,49),(8,31,26,50),(9,32,14,51),(10,33,15,52),(11,34,16,40),(12,35,17,41),(13,36,18,42)], [(27,46),(28,47),(29,48),(30,49),(31,50),(32,51),(33,52),(34,40),(35,41),(36,42),(37,43),(38,44),(39,45)], [(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)], [(1,18),(2,17),(3,16),(4,15),(5,14),(6,26),(7,25),(8,24),(9,23),(10,22),(11,21),(12,20),(13,19),(27,52),(28,51),(29,50),(30,49),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40)]])
D4×D13 is a maximal subgroup of
D52⋊1C4 D8⋊D13 Q8⋊D26 D4⋊6D26 D4⋊8D26
D4×D13 is a maximal quotient of
C22⋊Dic26 Dic13⋊4D4 C22⋊D52 D26.12D4 D26⋊D4 C23.6D26 C52⋊Q8 D52⋊8C4 D26.13D4 C4⋊2D52 D26⋊Q8 D8⋊D13 D8⋊3D13 Q8⋊D26 D4.D26 D26.6D4 Q16⋊D13 D104⋊C2 C23⋊D26 C52⋊2D4 Dic13⋊D4 C52⋊D4
40 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 13A | ··· | 13F | 26A | ··· | 26F | 26G | ··· | 26R | 52A | ··· | 52F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 13 | ··· | 13 | 26 | ··· | 26 | 26 | ··· | 26 | 52 | ··· | 52 |
| size | 1 | 1 | 2 | 2 | 13 | 13 | 26 | 26 | 2 | 26 | 2 | ··· | 2 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 |
40 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | D4 | D13 | D26 | D26 | D4×D13 |
| kernel | D4×D13 | C4×D13 | D52 | C13⋊D4 | D4×C13 | C22×D13 | D13 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 6 | 6 | 12 | 6 |
Matrix representation of D4×D13 ►in GL4(𝔽53) generated by
| 52 | 0 | 0 | 0 |
| 0 | 52 | 0 | 0 |
| 0 | 0 | 12 | 25 |
| 0 | 0 | 26 | 41 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 16 | 52 |
| 42 | 1 | 0 | 0 |
| 43 | 49 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 49 | 52 | 0 | 0 |
| 15 | 4 | 0 | 0 |
| 0 | 0 | 52 | 0 |
| 0 | 0 | 0 | 52 |
G:=sub<GL(4,GF(53))| [52,0,0,0,0,52,0,0,0,0,12,26,0,0,25,41],[1,0,0,0,0,1,0,0,0,0,1,16,0,0,0,52],[42,43,0,0,1,49,0,0,0,0,1,0,0,0,0,1],[49,15,0,0,52,4,0,0,0,0,52,0,0,0,0,52] >;
D4×D13 in GAP, Magma, Sage, TeX
D_4\times D_{13} % in TeX
G:=Group("D4xD13"); // GroupNames label
G:=SmallGroup(208,39);
// by ID
G=gap.SmallGroup(208,39);
# by ID
G:=PCGroup([5,-2,-2,-2,-2,-13,97,4804]);
// Polycyclic
G:=Group<a,b,c,d|a^4=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