Copied to
clipboard

## G = Dic5⋊2D5order 200 = 23·52

### The semidirect product of Dic5 and D5 acting through Inn(Dic5)

Aliases: Dic52D5, C10.2D10, C2.2D52, C5⋊D52C4, C52(C4×D5), C528(C2×C4), (C5×Dic5)⋊3C2, (C5×C10).2C22, (C2×C5⋊D5).1C2, SmallGroup(200,23)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — Dic5⋊2D5
 Chief series C1 — C5 — C52 — C5×C10 — C5×Dic5 — Dic5⋊2D5
 Lower central C52 — Dic5⋊2D5
 Upper central C1 — C2

Generators and relations for Dic52D5
G = < a,b,c,d | a10=c5=d2=1, b2=a5, bab-1=dad=a-1, ac=ca, bc=cb, bd=db, dcd=c-1 >

25C2
25C2
2C5
2C5
5C4
5C4
25C22
2C10
2C10
5D5
5D5
5D5
5D5
10D5
10D5
10D5
10D5
25C2×C4
5C20
5D10
5C20
5D10
10D10
10D10

Permutation representations of Dic52D5
On 20 points - transitive group 20T58
Generators in S20
(1 2 3 4 5 6 7 8 9 10)(11 12 13 14 15 16 17 18 19 20)
(1 12 6 17)(2 11 7 16)(3 20 8 15)(4 19 9 14)(5 18 10 13)
(1 3 5 7 9)(2 4 6 8 10)(11 19 17 15 13)(12 20 18 16 14)
(1 9)(2 8)(3 7)(4 6)(11 15)(12 14)(16 20)(17 19)

G:=sub<Sym(20)| (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12,6,17)(2,11,7,16)(3,20,8,15)(4,19,9,14)(5,18,10,13), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,9)(2,8)(3,7)(4,6)(11,15)(12,14)(16,20)(17,19)>;

G:=Group( (1,2,3,4,5,6,7,8,9,10)(11,12,13,14,15,16,17,18,19,20), (1,12,6,17)(2,11,7,16)(3,20,8,15)(4,19,9,14)(5,18,10,13), (1,3,5,7,9)(2,4,6,8,10)(11,19,17,15,13)(12,20,18,16,14), (1,9)(2,8)(3,7)(4,6)(11,15)(12,14)(16,20)(17,19) );

G=PermutationGroup([(1,2,3,4,5,6,7,8,9,10),(11,12,13,14,15,16,17,18,19,20)], [(1,12,6,17),(2,11,7,16),(3,20,8,15),(4,19,9,14),(5,18,10,13)], [(1,3,5,7,9),(2,4,6,8,10),(11,19,17,15,13),(12,20,18,16,14)], [(1,9),(2,8),(3,7),(4,6),(11,15),(12,14),(16,20),(17,19)])

G:=TransitiveGroup(20,58);

Dic52D5 is a maximal subgroup of
Dic5.4F5  Dic5.F5  C523C42  Dic5⋊F5  D52⋊C4  C2.D5≀C2  Dic10⋊D5  Dic105D5  C4×D52  Dic5.D10  D10⋊D10
Dic52D5 is a maximal quotient of
C20.29D10  C20.31D10  Dic52  C10.D20  C10.Dic10

32 conjugacy classes

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

32 irreducible representations

 dim 1 1 1 1 2 2 2 4 4 type + + + + + + + image C1 C2 C2 C4 D5 D10 C4×D5 D52 Dic5⋊2D5 kernel Dic5⋊2D5 C5×Dic5 C2×C5⋊D5 C5⋊D5 Dic5 C10 C5 C2 C1 # reps 1 2 1 4 4 4 8 4 4

Matrix representation of Dic52D5 in GL4(𝔽41) generated by

 35 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 32 0 0 0 28 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 1 0 0 0 6 40 0 0 0 0 6 40 0 0 35 35
G:=sub<GL(4,GF(41))| [35,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[32,28,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[1,6,0,0,0,40,0,0,0,0,6,35,0,0,40,35] >;

Dic52D5 in GAP, Magma, Sage, TeX

{\rm Dic}_5\rtimes_2D_5
% in TeX

G:=Group("Dic5:2D5");
// GroupNames label

G:=SmallGroup(200,23);
// by ID

G=gap.SmallGroup(200,23);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,20,26,328,4004]);
// Polycyclic

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

Export

׿
×
𝔽