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G = Dic5:2D5order 200 = 23·52

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

metabelian, supersoluble, monomial, A-group

Aliases: Dic5:2D5, C10.2D10, C2.2D52, C5:D5:2C4, C5:2(C4xD5), C52:8(C2xC4), (C5xDic5):3C2, (C5xC10).2C22, (C2xC5:D5).1C2, SmallGroup(200,23)

Series: Derived Chief Lower central Upper central

C1C52 — Dic5:2D5
C1C5C52C5xC10C5xDic5 — Dic5:2D5
C52 — Dic5:2D5
C1C2

Generators and relations for Dic5:2D5
 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 >

Subgroups: 244 in 42 conjugacy classes, 16 normal (6 characteristic)
Quotients: C1, C2, C4, C22, C2xC4, D5, D10, C4xD5, D52, Dic5:2D5
25C2
25C2
2C5
2C5
5C4
5C4
25C22
2C10
2C10
5D5
5D5
5D5
5D5
10D5
10D5
10D5
10D5
25C2xC4
5C20
5D10
5C20
5D10
10D10
10D10
5C4xD5
5C4xD5

Permutation representations of Dic5:2D5
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);

Dic5:2D5 is a maximal subgroup of
Dic5.4F5  Dic5.F5  C52:3C42  Dic5:F5  D52:C4  C2.D5wrC2  Dic10:D5  Dic10:5D5  C4xD52  Dic5.D10  D10:D10
Dic5:2D5 is a maximal quotient of
C20.29D10  C20.31D10  Dic52  C10.D20  C10.Dic10

32 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D5E5F5G5H10A10B10C10D10E10F10G10H20A···20H
order1222444455555555101010101010101020···20
size1125255555222244442222444410···10

32 irreducible representations

dim111122244
type+++++++
imageC1C2C2C4D5D10C4xD5D52Dic5:2D5
kernelDic5:2D5C5xDic5C2xC5:D5C5:D5Dic5C10C5C2C1
# reps121444844

Matrix representation of Dic5:2D5 in GL4(F41) generated by

35100
40000
0010
0001
,
32000
28900
0010
0001
,
1000
0100
00640
0010
,
1000
64000
00640
003535
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] >;

Dic5:2D5 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

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Subgroup lattice of Dic5:2D5 in TeX

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