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## G = D4×D13order 208 = 24·13

### Direct product of D4 and D13

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

 Derived series C1 — C26 — D4×D13
 Chief series C1 — C13 — C26 — D26 — C22×D13 — D4×D13
 Lower central C13 — C26 — D4×D13
 Upper central C1 — C2 — D4

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 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

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