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## G = Dic10⋊5D5order 400 = 24·52

### The semidirect product of Dic10 and D5 acting through Inn(Dic10)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C5×C10 — Dic10⋊5D5
 Chief series C1 — C5 — C52 — C5×C10 — D5×C10 — C5⋊D20 — Dic10⋊5D5
 Lower central C52 — C5×C10 — Dic10⋊5D5
 Upper central C1 — C2 — C4

Generators and relations for Dic105D5
G = < a,b,c,d | a20=c5=d2=1, b2=a10, bab-1=a-1, ac=ca, ad=da, bc=cb, dbd=a10b, dcd=c-1 >

Subgroups: 692 in 96 conjugacy classes, 32 normal (22 characteristic)
C1, C2, C2, C4, C4, C22, C5, C5, C2×C4, D4, Q8, D5, C10, C10, C4○D4, Dic5, Dic5, C20, C20, D10, D10, C2×C10, C52, Dic10, C4×D5, C4×D5, D20, C5⋊D4, C2×C20, C5×Q8, C5×D5, C5⋊D5, C5×C10, C4○D20, Q82D5, C5×Dic5, C5×Dic5, C5×C20, D5×C10, C2×C5⋊D5, Dic52D5, C5⋊D20, C5×Dic10, D5×C20, C20⋊D5, Dic105D5
Quotients: C1, C2, C22, C23, D5, C4○D4, D10, C22×D5, C4○D20, Q82D5, D52, C2×D52, Dic105D5

Smallest permutation representation of Dic105D5
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 25 11 35)(2 24 12 34)(3 23 13 33)(4 22 14 32)(5 21 15 31)(6 40 16 30)(7 39 17 29)(8 38 18 28)(9 37 19 27)(10 36 20 26)
(1 5 9 13 17)(2 6 10 14 18)(3 7 11 15 19)(4 8 12 16 20)(21 37 33 29 25)(22 38 34 30 26)(23 39 35 31 27)(24 40 36 32 28)
(1 39)(2 40)(3 21)(4 22)(5 23)(6 24)(7 25)(8 26)(9 27)(10 28)(11 29)(12 30)(13 31)(14 32)(15 33)(16 34)(17 35)(18 36)(19 37)(20 38)

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

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

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

52 conjugacy classes

 class 1 2A 2B 2C 2D 4A 4B 4C 4D 4E 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D 20E ··· 20N 20O 20P 20Q 20R 20S 20T 20U 20V 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 10 10 10 10 10 10 20 20 20 20 20 ··· 20 20 20 20 20 20 20 20 20 size 1 1 10 50 50 2 5 5 10 10 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 2 2 2 2 4 ··· 4 10 10 10 10 20 20 20 20

52 irreducible representations

 dim 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 D5 D5 C4○D4 D10 D10 D10 C4○D20 Q8⋊2D5 D52 C2×D52 Dic10⋊5D5 kernel Dic10⋊5D5 Dic5⋊2D5 C5⋊D20 C5×Dic10 D5×C20 C20⋊D5 Dic10 C4×D5 C52 Dic5 C20 D10 C5 C5 C4 C2 C1 # reps 1 2 2 1 1 1 2 2 2 6 4 2 8 2 4 4 8

Matrix representation of Dic105D5 in GL4(𝔽41) generated by

 14 30 0 0 11 9 0 0 0 0 1 0 0 0 0 1
,
 17 40 0 0 3 24 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 0 40 0 0 1 6
,
 17 40 0 0 1 24 0 0 0 0 1 6 0 0 0 40
G:=sub<GL(4,GF(41))| [14,11,0,0,30,9,0,0,0,0,1,0,0,0,0,1],[17,3,0,0,40,24,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,0,1,0,0,40,6],[17,1,0,0,40,24,0,0,0,0,1,0,0,0,6,40] >;

Dic105D5 in GAP, Magma, Sage, TeX

{\rm Dic}_{10}\rtimes_5D_5
% in TeX

G:=Group("Dic10:5D5");
// GroupNames label

G:=SmallGroup(400,168);
// by ID

G=gap.SmallGroup(400,168);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-5,-5,121,55,116,50,970,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^20=c^5=d^2=1,b^2=a^10,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d=a^10*b,d*c*d=c^-1>;
// generators/relations

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