C2xDic5:2D5 - GroupNames

G = C₂×Dic₅⋊₂D₅ order 400 = 2⁴·5²

Direct product of C₂ and Dic₅⋊₂D₅

direct product, metabelian, supersoluble, monomial, A-group

Aliases: C₂×Dic₅⋊₂D₅, Dic₅⋊₆D₁₀, C₁₀².₉C₂², C₁₀⋊₂(C₄×D₅), C₂².₉D₅², (C₂×Dic₅)⋊₅D₅, C₅²⋊₉(C₂²×C₄), (C₁₀×Dic₅)⋊₈C₂, (C₂×C₁₀).₁₃D₁₀, (C₅×C₁₀).₁₃C₂³, (C₅×Dic₅)⋊₇C₂², C₁₀.₁₃(C₂²×D₅), C₅⋊₃(C₂×C₄×D₅), C₂.₃(C₂×D₅²), C₅⋊D₅⋊₄(C₂×C₄), (C₂×C₅⋊D₅)⋊₅C₄, (C₅×C₁₀)⋊₈(C₂×C₄), (C₂²×C₅⋊D₅).₃C₂, (C₂×C₅⋊D₅).₁₇C₂², SmallGroup(400,175)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series

C₁ — C₅² — C₂×Dic₅⋊₂D₅

Chief series

C₁ — C₅ — C₅² — C₅×C₁₀ — C₅×Dic₅ — Dic₅⋊₂D₅ — C₂×Dic₅⋊₂D₅

Lower central

C₅² — C₂×Dic₅⋊₂D₅

Upper central

C₁ — C₂²

Generators and relations for C₂×Dic₅⋊₂D₅
G = < a,b,c,d,e | a²=b¹⁰=d⁵=e²=1, c²=b⁵, ab=ba, ac=ca, ad=da, ae=ea, cbc^-1=ebe=b^-1, bd=db, cd=dc, ce=ec, ede=d^-1 >

Subgroups: 908 in 140 conjugacy classes, 48 normal (8 characteristic)
C₁, C₂, C₂, C₂, C₄, C₂², C₂², C₅, C₅, C₂×C₄, C₂³, D₅, C₁₀, C₁₀, C₂²×C₄, Dic₅, C₂₀, D₁₀, C₂×C₁₀, C₂×C₁₀, C₅², C₄×D₅, C₂×Dic₅, C₂×C₂₀, C₂²×D₅, C₅⋊D₅, C₅×C₁₀, C₅×C₁₀, C₂×C₄×D₅, C₅×Dic₅, C₂×C₅⋊D₅, C₁₀², Dic₅⋊₂D₅, C₁₀×Dic₅, C₂²×C₅⋊D₅, C₂×Dic₅⋊₂D₅
Quotients: C₁, C₂, C₄, C₂², C₂×C₄, C₂³, D₅, C₂²×C₄, D₁₀, C₄×D₅, C₂²×D₅, C₂×C₄×D₅, D₅², Dic₅⋊₂D₅, C₂×D₅², C₂×Dic₅⋊₂D₅

Smallest permutation representation of C₂×Dic₅⋊₂D₅
►On 40 points

Generators in S₄₀

(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)(21 31)(22 32)(23 33)(24 34)(25 35)(26 36)(27 37)(28 38)(29 39)(30 40)

(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)

(1 24 6 29)(2 23 7 28)(3 22 8 27)(4 21 9 26)(5 30 10 25)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)

(1 5 9 3 7)(2 6 10 4 8)(11 15 19 13 17)(12 16 20 14 18)(21 27 23 29 25)(22 28 24 30 26)(31 37 33 39 35)(32 38 34 40 36)

(1 7)(2 6)(3 5)(8 10)(11 17)(12 16)(13 15)(18 20)(22 30)(23 29)(24 28)(25 27)(32 40)(33 39)(34 38)(35 37)

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

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

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

64 conjugacy classes

class	1	2A	2B	2C	2D	2E	2F	2G	4A	···	4H	5A	5B	5C	5D	5E	5F	5G	5H	10A	···	10L	10M	···	10X	20A	···	20P
order	1	2	2	2	2	2	2	2	4	···	4	5	5	5	5	5	5	5	5	10	···	10	10	···	10	20	···	20
size	1	1	1	1	25	25	25	25	5	···	5	2	2	2	2	4	4	4	4	2	···	2	4	···	4	10	···	10

64 irreducible representations

dim

type

image

C₁

C₂

C₄

D₅

D₁₀

C₄×D₅

D₅²

Dic₅⋊₂D₅

C₂×D₅²

kernel

C₂×Dic₅⋊₂D₅

Dic₅⋊₂D₅

C₁₀×Dic₅

C₂²×C₅⋊D₅

C₂×C₅⋊D₅

C₂×Dic₅

Dic₅

C₂×C₁₀

C₁₀

C₂²

C₂

# reps

Matrix representation of C₂×Dic₅⋊₂D₅ ►in GL₆(𝔽₄₁)

1	0	0	0	0	0
0	1	0	0	0	0
0	0	40	0	0	0
0	0	0	40	0	0
0	0	0	0	1	0
0	0	0	0	0	1

7	1	0	0	0	0
40	0	0	0	0	0
0	0	6	40	0	0
0	0	1	0	0	0
0	0	0	0	1	0
0	0	0	0	0	1

9	0	0	0	0	0
19	32	0	0	0	0
0	0	1	0	0	0
0	0	6	40	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	1	0	0	0
0	0	0	1	0	0
0	0	0	0	6	40
0	0	0	0	1	0

1	0	0	0	0	0
34	40	0	0	0	0
0	0	1	0	0	0
0	0	6	40	0	0
0	0	0	0	6	40
0	0	0	0	35	35

G:=sub<GL(6,GF(41))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[7,40,0,0,0,0,1,0,0,0,0,0,0,0,6,1,0,0,0,0,40,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,19,0,0,0,0,0,32,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,6,1,0,0,0,0,40,0],[1,34,0,0,0,0,0,40,0,0,0,0,0,0,1,6,0,0,0,0,0,40,0,0,0,0,0,0,6,35,0,0,0,0,40,35] >;

C₂×Dic₅⋊₂D₅ in GAP, Magma, Sage, TeX

C_2\times {\rm Dic}_5\rtimes_2D_5

% in TeX

G:=Group("C2xDic5:2D5");

// GroupNames label

G:=SmallGroup(400,175);

// by ID

G=gap.SmallGroup(400,175);

# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,48,55,970,11525]);

// Polycyclic

G:=Group<a,b,c,d,e|a^2=b^10=d^5=e^2=1,c^2=b^5,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c^-1=e*b*e=b^-1,b*d=d*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;

// generators/relations

G = C2×Dic5⋊2D5 order 400 = 24·52

Direct product of C2 and Dic5⋊2D5

G = C₂×Dic₅⋊₂D₅ order 400 = 2⁴·5²

Direct product of C₂ and Dic₅⋊₂D₅