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G = D4xD13order 208 = 24·13

Direct product of D4 and D13

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: D4xD13, C4:1D26, C52:C22, D52:3C2, C22:1D26, D26:2C22, C26.5C23, Dic13:1C22, C13:2(C2xD4), (C2xC26):C22, (C4xD13):1C2, (D4xC13):2C2, C13:D4:1C2, (C22xD13):2C2, C2.6(C22xD13), SmallGroup(208,39)

Series: Derived Chief Lower central Upper central

C1C26 — D4xD13
C1C13C26D26C22xD13 — D4xD13
C13C26 — D4xD13
C1C2D4

Generators and relations for D4xD13
 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, C2xC4, D4, D4, C23, C13, C2xD4, D13, D13, C26, C26, Dic13, C52, D26, D26, D26, C2xC26, C4xD13, D52, C13:D4, D4xC13, C22xD13, D4xD13
Quotients: C1, C2, C22, D4, C23, C2xD4, D13, D26, C22xD13, D4xD13

Smallest permutation representation of D4xD13
On 52 points
Generators in S52
(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)]])

D4xD13 is a maximal subgroup of
D52:1C4  D8:D13  Q8:D26  D4:6D26  D4:8D26
D4xD13 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 2A2B2C2D2E2F2G4A4B13A···13F26A···26F26G···26R52A···52F
order122222224413···1326···2626···2652···52
size1122131326262262···22···24···44···4

40 irreducible representations

dim11111122224
type+++++++++++
imageC1C2C2C2C2C2D4D13D26D26D4xD13
kernelD4xD13C4xD13D52C13:D4D4xC13C22xD13D13D4C4C22C1
# reps111212266126

Matrix representation of D4xD13 in GL4(F53) generated by

52000
05200
001225
002641
,
1000
0100
0010
001652
,
42100
434900
0010
0001
,
495200
15400
00520
00052
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] >;

D4xD13 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

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