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G = D4×D13order 208 = 24·13

Direct product of D4 and D13

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

Aliases: D4×D13, C41D26, C52⋊C22, D523C2, C221D26, D262C22, C26.5C23, Dic131C22, C132(C2×D4), (C2×C26)⋊C22, (C4×D13)⋊1C2, (D4×C13)⋊2C2, C13⋊D41C2, (C22×D13)⋊2C2, C2.6(C22×D13), SmallGroup(208,39)

Series: Derived Chief Lower central Upper central

C1C26 — D4×D13
C1C13C26D26C22×D13 — D4×D13
C13C26 — D4×D13
C1C2D4

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 [×6], C4, C4, C22 [×2], C22 [×7], C2×C4, D4, D4 [×3], C23 [×2], C13, C2×D4, D13 [×2], D13 [×2], C26, C26 [×2], Dic13, C52, D26, D26 [×2], D26 [×4], C2×C26 [×2], C4×D13, D52, C13⋊D4 [×2], D4×C13, C22×D13 [×2], D4×D13
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, C2×D4, D13, D26 [×3], C22×D13, D4×D13

Smallest permutation representation of D4×D13
On 52 points
Generators in S52
(1 36 23 49)(2 37 24 50)(3 38 25 51)(4 39 26 52)(5 27 14 40)(6 28 15 41)(7 29 16 42)(8 30 17 43)(9 31 18 44)(10 32 19 45)(11 33 20 46)(12 34 21 47)(13 35 22 48)
(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)
(1 22)(2 21)(3 20)(4 19)(5 18)(6 17)(7 16)(8 15)(9 14)(10 26)(11 25)(12 24)(13 23)(27 44)(28 43)(29 42)(30 41)(31 40)(32 52)(33 51)(34 50)(35 49)(36 48)(37 47)(38 46)(39 45)

G:=sub<Sym(52)| (1,36,23,49)(2,37,24,50)(3,38,25,51)(4,39,26,52)(5,27,14,40)(6,28,15,41)(7,29,16,42)(8,30,17,43)(9,31,18,44)(10,32,19,45)(11,33,20,46)(12,34,21,47)(13,35,22,48), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,44)(28,43)(29,42)(30,41)(31,40)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45)>;

G:=Group( (1,36,23,49)(2,37,24,50)(3,38,25,51)(4,39,26,52)(5,27,14,40)(6,28,15,41)(7,29,16,42)(8,30,17,43)(9,31,18,44)(10,32,19,45)(11,33,20,46)(12,34,21,47)(13,35,22,48), (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), (1,22)(2,21)(3,20)(4,19)(5,18)(6,17)(7,16)(8,15)(9,14)(10,26)(11,25)(12,24)(13,23)(27,44)(28,43)(29,42)(30,41)(31,40)(32,52)(33,51)(34,50)(35,49)(36,48)(37,47)(38,46)(39,45) );

G=PermutationGroup([(1,36,23,49),(2,37,24,50),(3,38,25,51),(4,39,26,52),(5,27,14,40),(6,28,15,41),(7,29,16,42),(8,30,17,43),(9,31,18,44),(10,32,19,45),(11,33,20,46),(12,34,21,47),(13,35,22,48)], [(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)], [(1,22),(2,21),(3,20),(4,19),(5,18),(6,17),(7,16),(8,15),(9,14),(10,26),(11,25),(12,24),(13,23),(27,44),(28,43),(29,42),(30,41),(31,40),(32,52),(33,51),(34,50),(35,49),(36,48),(37,47),(38,46),(39,45)])

D4×D13 is a maximal subgroup of
D521C4  D8⋊D13  Q8⋊D26  D46D26  D48D26
D4×D13 is a maximal quotient of
C22⋊Dic26  Dic134D4  C22⋊D52  D26.12D4  D26⋊D4  C23.6D26  C52⋊Q8  D528C4  D26.13D4  C42D52  D26⋊Q8  D8⋊D13  D83D13  Q8⋊D26  D4.D26  D26.6D4  Q16⋊D13  D104⋊C2  C23⋊D26  C522D4  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+++++++++++
imageC1C2C2C2C2C2D4D13D26D26D4×D13
kernelD4×D13C4×D13D52C13⋊D4D4×C13C22×D13D13D4C4C22C1
# reps111212266126

Matrix representation of D4×D13 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] >;

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

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