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G = Dic5.4F5  order 400 = 24·52

The non-split extension by Dic5 of F5 acting through Inn(Dic5)

metabelian, supersoluble, monomial, A-group

Aliases: Dic5.4F5, Dic5.6D10, C5⋊C8⋊3D5, C5⋊D5⋊1C8, C5⋊1(C8×D5), C52⋊3(C2×C8), C5⋊4(D5⋊C8), C2.3(D5×F5), C10.5(C4×D5), C52⋊3C8⋊2C2, C10.32(C2×F5), (C5×Dic5).3C4, Dic5⋊2D5.5C2, (C5×Dic5).7C22, (C5×C5⋊C8)⋊2C2, (C2×C5⋊D5).1C4, (C5×C10).5(C2×C4), SmallGroup(400,121)

Series: Derived ►Chief ►Lower central ►Upper central

Derived series
C1 — C52 — Dic5.4F5
Chief series
C1 — C5 — C52 — C5×C10 — C5×Dic5 — C5×C5⋊C8 — Dic5.4F5
Lower central
C52 — Dic5.4F5
Upper central
C1 — C2

Generators and relations for Dic5.4F5
 G = < a,b,c,d | a10=c5=1, b2=d4=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c3 >

C1
C2
25C2
25C2
C5
C5
4C5
5C4
5C4
25C22
C10
C10
4C10
5D5
5D5
5D5
5D5
20D5
20D5
C52
5C8
25C2×C4
25C8
Dic5
Dic5
5C20
5D10
5C20
5D10
20D10
C5⋊D5
C5⋊D5
C5×C10
25C2×C8
C5⋊C8
5C4×D5
5C5⋊C8
5C4×D5
5C40
5C5⋊2C8
C5×Dic5
C5×Dic5
C2×C5⋊D5
5C8×D5
5D5⋊C8
C52⋊3C8
Dic5⋊2D5
C5×C5⋊C8
Dic5.4F5

Smallest permutation representation of Dic5.4F5
►On 40 points
Generators in S40
(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 16 6 11)(2 15 7 20)(3 14 8 19)(4 13 9 18)(5 12 10 17)(21 36 26 31)(22 35 27 40)(23 34 28 39)(24 33 29 38)(25 32 30 37)

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

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

40 conjugacy classes

class 1 2A2B2C4A4B4C4D5A5B5C5D5E8A8B8C8D8E8F8G8H10A10B10C10D10E20A20B20C20D20E20F40A···40H
order122244445555588888888101010101020202020202040···40
size1125255555224885555252525252248810101010202010···10

40 irreducible representations

dim1111111222244488
type++++++++++
imageC1C2C2C2C4C4C8D5D10C4×D5C8×D5F5C2×F5D5⋊C8D5×F5Dic5.4F5
kernelDic5.4F5C5×C5⋊C8C52⋊3C8Dic5⋊2D5C5×Dic5C2×C5⋊D5C5⋊D5C5⋊C8Dic5C10C5Dic5C10C5C2C1
# reps1111228224811222

Matrix representation of Dic5.4F5 ►in GL6(𝔽41)

010000
4070000
001000
000100
000010
000001
,
090000
900000
001000
000100
000010
000001
,
100000
010000
0040100
0040001
0040000
0040010
,
3800000
0380000
000001
001000
000100
000010

G:=sub<GL(6,GF(41))| [0,40,0,0,0,0,1,7,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[0,9,0,0,0,0,9,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,40,40,40,40,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0],[38,0,0,0,0,0,0,38,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;
 

Dic5.4F5 in GAP, Magma, Sage, TeX 

{\rm Dic}_5._4F_5
 
% in TeX
 
G:=Group("Dic5.4F5");
 
// GroupNames label
 
G:=SmallGroup(400,121);
 
// by ID
 
G=gap.SmallGroup(400,121);
 
# by ID
 
G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,55,50,970,5765,2897]);
 
// Polycyclic
 
G:=Group<a,b,c,d|a^10=c^5=1,b^2=d^4=a^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^3>;
 
// generators/relations
 

Export

Subgroup lattice of Dic5.4F5 in TeX

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