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## G = C5×C5⋊D4order 200 = 23·52

### Direct product of C5 and C5⋊D4

Aliases: C5×C5⋊D4, C10C2, C526D4, Dic5⋊C10, D102C10, C1022C2, C10.21D10, C52(C5×D4), C22⋊(C5×D5), (C2×C10)⋊1D5, (C2×C10)⋊2C10, (D5×C10)⋊4C2, C2.5(D5×C10), C10.5(C2×C10), (C5×Dic5)⋊4C2, (C5×C10).10C22, SmallGroup(200,31)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C5⋊D4
 Chief series C1 — C5 — C10 — C5×C10 — D5×C10 — C5×C5⋊D4
 Lower central C5 — C10 — C5×C5⋊D4
 Upper central C1 — C10 — C2×C10

Generators and relations for C5×C5⋊D4
G = < a,b,c,d | a5=b5=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >

Permutation representations of C5×C5⋊D4
On 20 points - transitive group 20T53
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 17 7 12)(2 18 8 13)(3 19 9 14)(4 20 10 15)(5 16 6 11)
(1 12)(2 13)(3 14)(4 15)(5 11)(6 16)(7 17)(8 18)(9 19)(10 20)

G:=sub<Sym(20)| (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20)>;

G:=Group( (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20), (1,3,5,2,4)(6,8,10,7,9)(11,14,12,15,13)(16,19,17,20,18), (1,17,7,12)(2,18,8,13)(3,19,9,14)(4,20,10,15)(5,16,6,11), (1,12)(2,13)(3,14)(4,15)(5,11)(6,16)(7,17)(8,18)(9,19)(10,20) );

G=PermutationGroup([[(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)], [(1,3,5,2,4),(6,8,10,7,9),(11,14,12,15,13),(16,19,17,20,18)], [(1,17,7,12),(2,18,8,13),(3,19,9,14),(4,20,10,15),(5,16,6,11)], [(1,12),(2,13),(3,14),(4,15),(5,11),(6,16),(7,17),(8,18),(9,19),(10,20)]])

G:=TransitiveGroup(20,53);

C5×C5⋊D4 is a maximal subgroup of   Dic5.D10  D10.4D10  D10⋊D10  C5×D4×D5

65 conjugacy classes

 class 1 2A 2B 2C 4 5A 5B 5C 5D 5E ··· 5N 10A 10B 10C 10D 10E ··· 10AL 10AM 10AN 10AO 10AP 20A 20B 20C 20D order 1 2 2 2 4 5 5 5 5 5 ··· 5 10 10 10 10 10 ··· 10 10 10 10 10 20 20 20 20 size 1 1 2 10 10 1 1 1 1 2 ··· 2 1 1 1 1 2 ··· 2 10 10 10 10 10 10 10 10

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 type + + + + + + + image C1 C2 C2 C2 C5 C10 C10 C10 D4 D5 D10 C5⋊D4 C5×D4 C5×D5 D5×C10 C5×C5⋊D4 kernel C5×C5⋊D4 C5×Dic5 D5×C10 C102 C5⋊D4 Dic5 D10 C2×C10 C52 C2×C10 C10 C5 C5 C22 C2 C1 # reps 1 1 1 1 4 4 4 4 1 2 2 4 4 8 8 16

Matrix representation of C5×C5⋊D4 in GL2(𝔽11) generated by

 9 0 0 9
,
 8 1 2 10
,
 2 4 7 9
,
 1 1 0 10
G:=sub<GL(2,GF(11))| [9,0,0,9],[8,2,1,10],[2,7,4,9],[1,0,1,10] >;

C5×C5⋊D4 in GAP, Magma, Sage, TeX

C_5\times C_5\rtimes D_4
% in TeX

G:=Group("C5xC5:D4");
// GroupNames label

G:=SmallGroup(200,31);
// by ID

G=gap.SmallGroup(200,31);
# by ID

G:=PCGroup([5,-2,-2,-5,-2,-5,221,4004]);
// Polycyclic

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

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