D4xC13:C4 - GroupNames

G = D₄×C₁₃⋊C₄ order 416 = 2⁵·13

Direct product of D₄ and C₁₃⋊C₄

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

Aliases: D₄×C₁₃⋊C₄, D₅₂⋊₃C₄, D₂₆.₁₂C₂³, C₁₃⋊(C₄×D₄), C₅₂⋊(C₂×C₄), D₂₆⋊(C₂×C₄), C₁₃⋊D₄⋊C₄, (D₄×C₁₃)⋊₃C₄, Dic₁₃⋊(C₂×C₄), C₅₂⋊C₄⋊₂C₂, D₁₃.₂(C₂×D₄), (D₄×D₁₃).₃C₂, D₁₃.D₄⋊₃C₂, C₂₆.₈(C₂²×C₄), D₁₃.₂(C₄○D₄), (C₄×D₁₃).₁₂C₂², (C₂²×D₁₃).₁₆C₂², C₄⋊₁(C₂×C₁₃⋊C₄), (C₂×C₂₆)⋊(C₂×C₄), (C₄×C₁₃⋊C₄)⋊₃C₂, C₂²⋊₁(C₂×C₁₃⋊C₄), (C₂²×C₁₃⋊C₄)⋊₁C₂, C₂.₉(C₂²×C₁₃⋊C₄), (C₂×C₁₃⋊C₄).₃C₂², SmallGroup(416,206)

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

type

image

C₁

C₂

C₄

D₄

C₄○D₄

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

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

G = D4×C13⋊C4 order 416 = 25·13

Direct product of D4 and C13⋊C4

G = D₄×C₁₃⋊C₄ order 416 = 2⁵·13

Direct product of D₄ and C₁₃⋊C₄