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## G = D4×C13⋊C4order 416 = 25·13

### Direct product of D4 and C13⋊C4

Series: Derived Chief Lower central Upper central

 Derived series C1 — C26 — D4×C13⋊C4
 Chief series C1 — C13 — D13 — D26 — C2×C13⋊C4 — C22×C13⋊C4 — D4×C13⋊C4
 Lower central C13 — C26 — D4×C13⋊C4
 Upper central C1 — C2 — D4

Generators and relations for D4×C13⋊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, C2×C4, D4, D4, C23, C13, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, D13, D13, C26, C26, C4×D4, Dic13, C52, C13⋊C4, C13⋊C4, D26, D26, D26, C2×C26, C4×D13, D52, C13⋊D4, D4×C13, C2×C13⋊C4, C2×C13⋊C4, C2×C13⋊C4, C22×D13, C4×C13⋊C4, C52⋊C4, D13.D4, D4×D13, C22×C13⋊C4, D4×C13⋊C4
Quotients: C1, C2, C4, C22, C2×C4, D4, C23, C22×C4, C2×D4, C4○D4, C4×D4, C13⋊C4, C2×C13⋊C4, C22×C13⋊C4, D4×C13⋊C4

Smallest permutation representation of D4×C13⋊C4
On 52 points
Generators in S52
(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 C4○D4 C13⋊C4 C2×C13⋊C4 C2×C13⋊C4 D4×C13⋊C4 kernel D4×C13⋊C4 C4×C13⋊C4 C52⋊C4 D13.D4 D4×D13 C22×C13⋊C4 D52 C13⋊D4 D4×C13 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 D4×C13⋊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] >;

D4×C13⋊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

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