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## G = Dic5.D10order 400 = 24·52

### 3rd non-split extension by Dic5 of D10 acting via D10/D5=C2

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — Dic5.D10
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — D5×Dic5 — Dic5.D10
 Lower central C52 — C5×C10 — Dic5.D10
 Upper central C1 — C2 — C22

Generators and relations for Dic5.D10
G = < a,b,c,d | a10=c10=1, b2=d2=a5, bab-1=cac-1=dad-1=a-1, cbc-1=dbd-1=a5b, dcd-1=c-1 >

Subgroups: 596 in 96 conjugacy classes, 32 normal (all characteristic)
C1, C2, C2 [×3], C4 [×4], C22, C22 [×2], C5 [×2], C5 [×2], C2×C4 [×3], D4 [×3], Q8, D5 [×5], C10 [×2], C10 [×7], C4○D4, Dic5 [×3], Dic5 [×4], C20 [×3], D10, D10 [×4], C2×C10 [×2], C2×C10 [×3], C52, Dic10 [×2], C4×D5 [×3], D20, C2×Dic5, C2×Dic5, C5⋊D4, C5⋊D4 [×5], C2×C20, C5×D4, C5×D5, C5⋊D5, C5×C10, C5×C10, C4○D20, D42D5, C5×Dic5 [×3], C526C4, D5×C10, C2×C5⋊D5, C102, D5×Dic5, Dic52D5, C5⋊D20, C522Q8, C10×Dic5, C5×C5⋊D4, C527D4, Dic5.D10
Quotients: C1, C2 [×7], C22 [×7], C23, D5 [×2], C4○D4, D10 [×6], C22×D5 [×2], C4○D20, D42D5, D52, C2×D52, Dic5.D10

Smallest permutation representation of Dic5.D10
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 26 6 21)(2 25 7 30)(3 24 8 29)(4 23 9 28)(5 22 10 27)(11 34 16 39)(12 33 17 38)(13 32 18 37)(14 31 19 36)(15 40 20 35)
(1 29 5 25 9 21 3 27 7 23)(2 28 6 24 10 30 4 26 8 22)(11 31 17 35 13 39 19 33 15 37)(12 40 18 34 14 38 20 32 16 36)
(1 34 6 39)(2 33 7 38)(3 32 8 37)(4 31 9 36)(5 40 10 35)(11 21 16 26)(12 30 17 25)(13 29 18 24)(14 28 19 23)(15 27 20 22)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E 5F 5G 5H 10A ··· 10H 10I ··· 10V 10W 10X 20A ··· 20H 20I 20J order 1 2 2 2 2 4 4 4 4 4 5 5 5 5 5 5 5 5 10 ··· 10 10 ··· 10 10 10 20 ··· 20 20 20 size 1 1 2 10 50 5 5 10 10 50 2 2 2 2 4 4 4 4 2 ··· 2 4 ··· 4 20 20 10 ··· 10 20 20

52 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 4 4 4 4 type + + + + + + + + + + + + + - + + image C1 C2 C2 C2 C2 C2 C2 C2 D5 D5 C4○D4 D10 D10 D10 C4○D20 D4⋊2D5 D52 C2×D52 Dic5.D10 kernel Dic5.D10 D5×Dic5 Dic5⋊2D5 C5⋊D20 C52⋊2Q8 C10×Dic5 C5×C5⋊D4 C52⋊7D4 C2×Dic5 C5⋊D4 C52 Dic5 D10 C2×C10 C5 C5 C22 C2 C1 # reps 1 1 1 1 1 1 1 1 2 2 2 6 2 4 8 2 4 4 8

Matrix representation of Dic5.D10 in GL6(𝔽41)

 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 34 40 0 0 0 0 1 0
,
 12 23 0 0 0 0 24 29 0 0 0 0 0 0 40 0 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40
,
 12 23 0 0 0 0 33 29 0 0 0 0 0 0 40 34 0 0 0 0 7 7 0 0 0 0 0 0 1 0 0 0 0 0 34 40
,
 15 39 0 0 0 0 31 26 0 0 0 0 0 0 1 7 0 0 0 0 0 40 0 0 0 0 0 0 1 0 0 0 0 0 34 40

G:=sub<GL(6,GF(41))| [40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,34,1,0,0,0,0,40,0],[12,24,0,0,0,0,23,29,0,0,0,0,0,0,40,0,0,0,0,0,0,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[12,33,0,0,0,0,23,29,0,0,0,0,0,0,40,7,0,0,0,0,34,7,0,0,0,0,0,0,1,34,0,0,0,0,0,40],[15,31,0,0,0,0,39,26,0,0,0,0,0,0,1,0,0,0,0,0,7,40,0,0,0,0,0,0,1,34,0,0,0,0,0,40] >;

Dic5.D10 in GAP, Magma, Sage, TeX

{\rm Dic}_5.D_{10}
% in TeX

G:=Group("Dic5.D10");
// GroupNames label

G:=SmallGroup(400,173);
// by ID

G=gap.SmallGroup(400,173);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,55,218,970,11525]);
// Polycyclic

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

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