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## G = D4⋊2Dic5order 160 = 25·5

### 2nd semidirect product of D4 and Dic5 acting via Dic5/C10=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C20 — D4⋊2Dic5
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C4.Dic5 — D4⋊2Dic5
 Lower central C5 — C10 — C20 — D4⋊2Dic5
 Upper central C1 — C4 — C2×C4 — C4○D4

Generators and relations for D42Dic5
G = < a,b,c,d | a4=c10=1, b2=a2, d2=c5, bab-1=a-1, ac=ca, ad=da, cbc-1=a2b, dbd-1=a-1b, dcd-1=c-1 >

Smallest permutation representation of D42Dic5
On 40 points
Generators in S40
(1 20 14 6)(2 16 15 7)(3 17 11 8)(4 18 12 9)(5 19 13 10)(21 31 26 36)(22 32 27 37)(23 33 28 38)(24 34 29 39)(25 35 30 40)
(1 31 14 36)(2 37 15 32)(3 33 11 38)(4 39 12 34)(5 35 13 40)(6 26 20 21)(7 22 16 27)(8 28 17 23)(9 24 18 29)(10 30 19 25)
(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 12)(2 11)(3 15)(4 14)(5 13)(6 18)(7 17)(8 16)(9 20)(10 19)(21 39 26 34)(22 38 27 33)(23 37 28 32)(24 36 29 31)(25 35 30 40)

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

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

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

34 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 4E 4F 4G 4H 5A 5B 8A 8B 10A 10B 10C ··· 10H 20A 20B 20C 20D 20E ··· 20J order 1 2 2 2 4 4 4 4 4 4 4 4 5 5 8 8 10 10 10 ··· 10 20 20 20 20 20 ··· 20 size 1 1 2 4 1 1 2 4 10 10 10 10 2 2 20 20 2 2 4 ··· 4 2 2 2 2 4 ··· 4

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 4 type + + + + + + + + - - image C1 C2 C2 C2 C4 C4 D4 D4 D5 D10 Dic5 Dic5 C4≀C2 C5⋊D4 C5⋊D4 D4⋊2Dic5 kernel D4⋊2Dic5 C4.Dic5 C4×Dic5 C5×C4○D4 C5×D4 C5×Q8 C20 C2×C10 C4○D4 C2×C4 D4 Q8 C5 C4 C22 C1 # reps 1 1 1 1 2 2 1 1 2 2 2 2 4 4 4 4

Matrix representation of D42Dic5 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 32 0 0 0 0 9
,
 17 1 0 0 40 24 0 0 0 0 0 32 0 0 32 0
,
 34 1 0 0 40 0 0 0 0 0 40 0 0 0 0 1
,
 7 34 0 0 1 34 0 0 0 0 32 0 0 0 0 40
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,32,0,0,0,0,9],[17,40,0,0,1,24,0,0,0,0,0,32,0,0,32,0],[34,40,0,0,1,0,0,0,0,0,40,0,0,0,0,1],[7,1,0,0,34,34,0,0,0,0,32,0,0,0,0,40] >;

D42Dic5 in GAP, Magma, Sage, TeX

D_4\rtimes_2{\rm Dic}_5
% in TeX

G:=Group("D4:2Dic5");
// GroupNames label

G:=SmallGroup(160,44);
// by ID

G=gap.SmallGroup(160,44);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,24,121,86,579,297,69,4613]);
// Polycyclic

G:=Group<a,b,c,d|a^4=c^10=1,b^2=a^2,d^2=c^5,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^2*b,d*b*d^-1=a^-1*b,d*c*d^-1=c^-1>;
// generators/relations

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