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## G = C5×C52⋊C4order 500 = 22·53

### Direct product of C5 and C52⋊C4

Aliases: C5×C52⋊C4, C534C4, C526F5, C527C20, C52(C5×F5), C5⋊D5.3C10, (C5×C5⋊D5).2C2, SmallGroup(500,44)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5×C52⋊C4
 Chief series C1 — C5 — C52 — C5⋊D5 — C5×C5⋊D5 — C5×C52⋊C4
 Lower central C52 — C5×C52⋊C4
 Upper central C1 — C5

Generators and relations for C5×C52⋊C4
G = < a,b,c,d | a5=b5=c5=d4=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b2, dcd-1=c3 >

25C2
2C5
2C5
4C5
4C5
4C5
4C5
4C5
4C5
25C4
5D5
5D5
10D5
10D5
25C10
2C52
2C52
4C52
4C52
4C52
4C52
4C52
4C52
5F5
5F5
25C20
10C5×D5
10C5×D5

Permutation representations of C5×C52⋊C4
On 20 points - transitive group 20T126
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 9 7 10 8)(11 12 13 14 15)(16 20 19 18 17)
(1 2 3 4 5)(6 10 9 8 7)(11 13 15 12 14)(16 19 17 20 18)
(1 18 8 13)(2 19 9 14)(3 20 10 15)(4 16 6 11)(5 17 7 12)

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

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

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

G:=TransitiveGroup(20,126);

50 conjugacy classes

 class 1 2 4A 4B 5A 5B 5C 5D 5E ··· 5AH 10A 10B 10C 10D 20A ··· 20H order 1 2 4 4 5 5 5 5 5 ··· 5 10 10 10 10 20 ··· 20 size 1 25 25 25 1 1 1 1 4 ··· 4 25 25 25 25 25 ··· 25

50 irreducible representations

 dim 1 1 1 1 1 1 4 4 4 4 type + + + + image C1 C2 C4 C5 C10 C20 F5 C5×F5 C52⋊C4 C5×C52⋊C4 kernel C5×C52⋊C4 C5×C5⋊D5 C53 C52⋊C4 C5⋊D5 C52 C52 C5 C5 C1 # reps 1 1 2 4 4 8 2 8 4 16

Matrix representation of C5×C52⋊C4 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 37 0 0 0 0 10 0 0 0 0 18 0 0 0 0 16
,
 18 0 0 0 0 16 0 0 0 0 37 0 0 0 0 10
,
 0 0 1 0 0 0 0 1 0 1 0 0 1 0 0 0
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[37,0,0,0,0,10,0,0,0,0,18,0,0,0,0,16],[18,0,0,0,0,16,0,0,0,0,37,0,0,0,0,10],[0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,0] >;

C5×C52⋊C4 in GAP, Magma, Sage, TeX

C_5\times C_5^2\rtimes C_4
% in TeX

G:=Group("C5xC5^2:C4");
// GroupNames label

G:=SmallGroup(500,44);
// by ID

G=gap.SmallGroup(500,44);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,1203,173,5004,1014]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^5=c^5=d^4=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^2,d*c*d^-1=c^3>;
// generators/relations

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