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## G = D5×Dic5order 200 = 23·52

### Direct product of D5 and Dic5

Aliases: D5×Dic5, D10.2D5, C10.1D10, C2.1D52, C54(C4×D5), (C5×D5)⋊4C4, C527(C2×C4), C52(C2×Dic5), C526C41C2, (C5×Dic5)⋊2C2, (D5×C10).1C2, (C5×C10).1C22, SmallGroup(200,22)

Series: Derived Chief Lower central Upper central

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

Generators and relations for D5×Dic5
G = < a,b,c,d | a5=b2=c10=1, d2=c5, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Smallest permutation representation of D5×Dic5
On 40 points
Generators in S40
(1 5 9 3 7)(2 6 10 4 8)(11 17 13 19 15)(12 18 14 20 16)(21 27 23 29 25)(22 28 24 30 26)(31 35 39 33 37)(32 36 40 34 38)
(1 25)(2 26)(3 27)(4 28)(5 29)(6 30)(7 21)(8 22)(9 23)(10 24)(11 37)(12 38)(13 39)(14 40)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)
(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 14 6 19)(2 13 7 18)(3 12 8 17)(4 11 9 16)(5 20 10 15)(21 34 26 39)(22 33 27 38)(23 32 28 37)(24 31 29 36)(25 40 30 35)

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

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

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

D5×Dic5 is a maximal subgroup of
D5.Dic10  D10.F5  D10.2F5  C524M4(2)  D205D5  D20⋊D5  C4×D52  Dic5.D10  D10.4D10
D5×Dic5 is a maximal quotient of
C20.30D10  D10⋊Dic5  Dic5⋊Dic5

32 conjugacy classes

 class 1 2A 2B 2C 4A 4B 4C 4D 5A 5B 5C 5D 5E 5F 5G 5H 10A 10B 10C 10D 10E 10F 10G 10H 10I 10J 10K 10L 20A 20B 20C 20D order 1 2 2 2 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 size 1 1 5 5 5 5 25 25 2 2 2 2 4 4 4 4 2 2 2 2 4 4 4 4 10 10 10 10 10 10 10 10

32 irreducible representations

 dim 1 1 1 1 1 2 2 2 2 2 4 4 type + + + + + + - + + - image C1 C2 C2 C2 C4 D5 D5 Dic5 D10 C4×D5 D52 D5×Dic5 kernel D5×Dic5 C5×Dic5 C52⋊6C4 D5×C10 C5×D5 Dic5 D10 D5 C10 C5 C2 C1 # reps 1 1 1 1 4 2 2 4 4 4 4 4

Matrix representation of D5×Dic5 in GL4(𝔽41) generated by

 1 0 0 0 0 1 0 0 0 0 6 40 0 0 1 0
,
 40 0 0 0 0 40 0 0 0 0 6 40 0 0 35 35
,
 35 1 0 0 40 0 0 0 0 0 1 0 0 0 0 1
,
 9 0 0 0 13 32 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(41))| [1,0,0,0,0,1,0,0,0,0,6,1,0,0,40,0],[40,0,0,0,0,40,0,0,0,0,6,35,0,0,40,35],[35,40,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[9,13,0,0,0,32,0,0,0,0,1,0,0,0,0,1] >;

D5×Dic5 in GAP, Magma, Sage, TeX

D_5\times {\rm Dic}_5
% in TeX

G:=Group("D5xDic5");
// GroupNames label

G:=SmallGroup(200,22);
// by ID

G=gap.SmallGroup(200,22);
# by ID

G:=PCGroup([5,-2,-2,-2,-5,-5,26,328,4004]);
// Polycyclic

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

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