direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4xC13:C4, D52:3C4, D26.12C23, C13:(C4xD4), C52:(C2xC4), D26:(C2xC4), C13:D4:C4, (D4xC13):3C4, Dic13:(C2xC4), C52:C4:2C2, D13.2(C2xD4), (D4xD13).3C2, D13.D4:3C2, C26.8(C22xC4), D13.2(C4oD4), (C4xD13).12C22, (C22xD13).16C22, C4:1(C2xC13:C4), (C2xC26):(C2xC4), (C4xC13:C4):3C2, C22:1(C2xC13:C4), (C22xC13:C4):1C2, C2.9(C22xC13:C4), (C2xC13:C4).3C22, SmallGroup(416,206)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4xC13:C4
G = < a,b,c,d | a4=b2=c13=d4=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c5 >
Subgroups: 748 in 94 conjugacy classes, 38 normal (20 characteristic)
C1, C2, C2, C4, C4, C22, C22, C2xC4, D4, D4, C23, C13, C42, C22:C4, C4:C4, C22xC4, C2xD4, D13, D13, C26, C26, C4xD4, Dic13, C52, C13:C4, C13:C4, D26, D26, D26, C2xC26, C4xD13, D52, C13:D4, D4xC13, C2xC13:C4, C2xC13:C4, C2xC13:C4, C22xD13, C4xC13:C4, C52:C4, D13.D4, D4xD13, C22xC13:C4, D4xC13:C4
Quotients: C1, C2, C4, C22, C2xC4, D4, C23, C22xC4, C2xD4, C4oD4, C4xD4, C13:C4, C2xC13:C4, C22xC13:C4, D4xC13:C4
►On 52 points
(1 27 14 40)(2 28 15 41)(3 29 16 42)(4 30 17 43)(5 31 18 44)(6 32 19 45)(7 33 20 46)(8 34 21 47)(9 35 22 48)(10 36 23 49)(11 37 24 50)(12 38 25 51)(13 39 26 52)
(27 40)(28 41)(29 42)(30 43)(31 44)(32 45)(33 46)(34 47)(35 48)(36 49)(37 50)(38 51)(39 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)
(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)
G:=sub<Sym(52)| (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (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)>;
G:=Group( (1,27,14,40)(2,28,15,41)(3,29,16,42)(4,30,17,43)(5,31,18,44)(6,32,19,45)(7,33,20,46)(8,34,21,47)(9,35,22,48)(10,36,23,49)(11,37,24,50)(12,38,25,51)(13,39,26,52), (27,40)(28,41)(29,42)(30,43)(31,44)(32,45)(33,46)(34,47)(35,48)(36,49)(37,50)(38,51)(39,52), (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), (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) );
G=PermutationGroup([[(1,27,14,40),(2,28,15,41),(3,29,16,42),(4,30,17,43),(5,31,18,44),(6,32,19,45),(7,33,20,46),(8,34,21,47),(9,35,22,48),(10,36,23,49),(11,37,24,50),(12,38,25,51),(13,39,26,52)], [(27,40),(28,41),(29,42),(30,43),(31,44),(32,45),(33,46),(34,47),(35,48),(36,49),(37,50),(38,51),(39,52)], [(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)], [(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)]])
35 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 4A | 4B | 4C | 4D | 4E | 4F | ··· | 4L | 13A | 13B | 13C | 26A | 26B | 26C | 26D | ··· | 26I | 52A | 52B | 52C |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 13 | 13 | 13 | 26 | 26 | 26 | 26 | ··· | 26 | 52 | 52 | 52 |
| size | 1 | 1 | 2 | 2 | 13 | 13 | 26 | 26 | 2 | 13 | 13 | 13 | 13 | 26 | ··· | 26 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 |
35 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | D4 | C4oD4 | C13:C4 | C2xC13:C4 | C2xC13:C4 | D4xC13:C4 |
| kernel | D4xC13:C4 | C4xC13:C4 | C52:C4 | D13.D4 | D4xD13 | C22xC13:C4 | D52 | C13:D4 | D4xC13 | C13:C4 | D13 | D4 | C4 | C22 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 2 | 2 | 3 | 3 | 6 | 3 |
Matrix representation of D4xC13:C4 ►in GL6(F53)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 52 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 41 | 45 | 7 | 52 |
| 0 | 0 | 51 | 51 | 15 | 13 |
| 0 | 0 | 29 | 38 | 14 | 51 |
| 0 | 0 | 28 | 1 | 20 | 20 |
| 30 | 0 | 0 | 0 | 0 | 0 |
| 0 | 30 | 0 | 0 | 0 | 0 |
| 0 | 0 | 17 | 47 | 20 | 44 |
| 0 | 0 | 3 | 40 | 11 | 21 |
| 0 | 0 | 9 | 42 | 11 | 38 |
| 0 | 0 | 20 | 32 | 27 | 38 |
G:=sub<GL(6,GF(53))| [0,52,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[52,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,41,51,29,28,0,0,45,51,38,1,0,0,7,15,14,20,0,0,52,13,51,20],[30,0,0,0,0,0,0,30,0,0,0,0,0,0,17,3,9,20,0,0,47,40,42,32,0,0,20,11,11,27,0,0,44,21,38,38] >;
D4xC13:C4 in GAP, Magma, Sage, TeX
D_4\times C_{13}\rtimes C_4 % in TeX
G:=Group("D4xC13:C4"); // GroupNames label
G:=SmallGroup(416,206);
// by ID
G=gap.SmallGroup(416,206);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-2,-13,48,188,9221,1751]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=c^13=d^4=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^-1=c^5>;
// generators/relations