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## G = Dic5.F5order 400 = 24·52

### 1st non-split extension by Dic5 of F5 acting via F5/D5=C2

Series: Derived Chief Lower central Upper central

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

Generators and relations for Dic5.F5
G = < a,b,c,d | a10=c5=1, b2=d4=a5, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd-1=a5b, dcd-1=c3 >

Smallest permutation representation of Dic5.F5
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 11 6 16)(2 20 7 15)(3 19 8 14)(4 18 9 13)(5 17 10 12)(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), (1,11,6,16)(2,20,7,15)(3,19,8,14)(4,18,9,13)(5,17,10,12)(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), (1,11,6,16)(2,20,7,15)(3,19,8,14)(4,18,9,13)(5,17,10,12)(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)], [(1,11,6,16),(2,20,7,15),(3,19,8,14),(4,18,9,13),(5,17,10,12),(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)]])

34 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E 8A 8B 8C 8D 10A 10B 10C 10D 10E 20A 20B 20C 20D 20E 20F 40A ··· 40H order 1 2 2 4 4 4 5 5 5 5 5 8 8 8 8 10 10 10 10 10 20 20 20 20 20 20 40 ··· 40 size 1 1 50 5 5 10 2 2 4 8 8 10 10 50 50 2 2 4 8 8 10 10 10 10 20 20 10 ··· 10

34 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 2 4 4 4 8 8 type + + + + + + + + + + image C1 C2 C2 C2 C4 C4 D5 M4(2) D10 C4×D5 C8⋊D5 F5 C2×F5 C4.F5 D5×F5 Dic5.F5 kernel Dic5.F5 C5×C5⋊C8 C52⋊3C8 Dic5⋊2D5 C5×Dic5 C2×C5⋊D5 C5⋊C8 C52 Dic5 C10 C5 Dic5 C10 C5 C2 C1 # reps 1 1 1 1 2 2 2 2 2 4 8 1 1 2 2 2

Matrix representation of Dic5.F5 in GL6(𝔽41)

 35 1 0 0 0 0 40 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
,
 32 0 0 0 0 0 28 9 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 1 0 0 0 0 40 0 0 1 0 0 40 0 0 0 0 0 40 0 1 0
,
 13 23 0 0 0 0 18 28 0 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0

G:=sub<GL(6,GF(41))| [35,40,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[32,28,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[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],[13,18,0,0,0,0,23,28,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0] >;

Dic5.F5 in GAP, Magma, Sage, TeX

{\rm Dic}_5.F_5
% in TeX

G:=Group("Dic5.F5");
// GroupNames label

G:=SmallGroup(400,123);
// by ID

G=gap.SmallGroup(400,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,24,121,55,50,970,5765,2897]);
// 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,d*b*d^-1=a^5*b,d*c*d^-1=c^3>;
// generators/relations

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