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## G = C5×D52order 500 = 22·53

### Direct product of C5, D5 and D5

Aliases: C5×D52, C526D10, C531C22, (C5×D5)⋊C10, C5⋊D52C10, C51(D5×C10), C523(C2×C10), (D5×C52)⋊1C2, (C5×C5⋊D5)⋊1C2, SmallGroup(500,50)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C52 — C5×D52
 Chief series C1 — C5 — C52 — C53 — D5×C52 — C5×D52
 Lower central C52 — C5×D52
 Upper central C1 — C5

Generators and relations for C5×D52
G = < a,b,c,d,e | a5=b5=c2=d5=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >

Subgroups: 344 in 72 conjugacy classes, 20 normal (8 characteristic)
C1, C2, C22, C5, C5, C5, D5, D5, C10, D10, C2×C10, C52, C52, C52, C5×D5, C5×D5, C5⋊D5, C5×C10, D52, D5×C10, C53, D5×C52, C5×C5⋊D5, C5×D52
Quotients: C1, C2, C22, C5, D5, C10, D10, C2×C10, C5×D5, D52, D5×C10, C5×D52

Permutation representations of C5×D52
On 20 points - transitive group 20T124
Generators in S20
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 2 3 4 5)(6 10 9 8 7)(11 12 13 14 15)(16 20 19 18 17)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 5 4 3 2)(6 10 9 8 7)(11 12 13 14 15)(16 17 18 19 20)
(1 15)(2 11)(3 12)(4 13)(5 14)(6 20)(7 16)(8 17)(9 18)(10 19)

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

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

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

G:=TransitiveGroup(20,124);

80 conjugacy classes

 class 1 2A 2B 2C 5A 5B 5C 5D 5E ··· 5X 5Y ··· 5AR 10A ··· 10H 10I ··· 10AB 10AC 10AD 10AE 10AF order 1 2 2 2 5 5 5 5 5 ··· 5 5 ··· 5 10 ··· 10 10 ··· 10 10 10 10 10 size 1 5 5 25 1 1 1 1 2 ··· 2 4 ··· 4 5 ··· 5 10 ··· 10 25 25 25 25

80 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 type + + + + + + image C1 C2 C2 C5 C10 C10 D5 D10 C5×D5 D5×C10 D52 C5×D52 kernel C5×D52 D5×C52 C5×C5⋊D5 D52 C5×D5 C5⋊D5 C5×D5 C52 D5 C5 C5 C1 # reps 1 2 1 4 8 4 4 4 16 16 4 16

Matrix representation of C5×D52 in GL4(𝔽11) generated by

 9 0 0 0 0 9 0 0 0 0 1 0 0 0 0 1
,
 1 0 0 0 0 1 0 0 0 0 3 10 0 0 1 0
,
 1 0 0 0 0 1 0 0 0 0 3 10 0 0 8 8
,
 0 10 0 0 1 3 0 0 0 0 1 0 0 0 0 1
,
 1 3 0 0 0 10 0 0 0 0 1 0 0 0 0 1
G:=sub<GL(4,GF(11))| [9,0,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,0,1,0,0,0,0,3,1,0,0,10,0],[1,0,0,0,0,1,0,0,0,0,3,8,0,0,10,8],[0,1,0,0,10,3,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,3,10,0,0,0,0,1,0,0,0,0,1] >;

C5×D52 in GAP, Magma, Sage, TeX

C_5\times D_5^2
% in TeX

G:=Group("C5xD5^2");
// GroupNames label

G:=SmallGroup(500,50);
// by ID

G=gap.SmallGroup(500,50);
# by ID

G:=PCGroup([5,-2,-2,-5,-5,-5,808,10004]);
// Polycyclic

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

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