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## G = C5×D5.D5order 500 = 22·53

### Direct product of C5 and D5.D5

Aliases: C5×D5.D5, C532C4, C525C20, C5211F5, C524Dic5, D5.(C5×D5), C5⋊(C5×Dic5), C53(C5×F5), (C5×D5).1D5, (C5×D5).3C10, (D5×C52).2C2, SmallGroup(500,42)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C5×D5.D5
G = < a,b,c,d,e | a5=b5=c2=d5=1, e2=b-1c, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, ebe-1=b2, cd=dc, ece-1=bc, ede-1=d-1 >

Permutation representations of C5×D5.D5
On 20 points - transitive group 20T127
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 9)(2 10)(3 6)(4 7)(5 8)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 3 5 2 4)(6 8 10 7 9)(11 14 12 15 13)(16 19 17 20 18)
(1 19 7 14)(2 20 8 15)(3 16 9 11)(4 17 10 12)(5 18 6 13)

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

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

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

G:=TransitiveGroup(20,127);

65 conjugacy classes

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

65 irreducible representations

 dim 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + - + image C1 C2 C4 C5 C10 C20 D5 Dic5 C5×D5 C5×Dic5 F5 C5×F5 D5.D5 C5×D5.D5 kernel C5×D5.D5 D5×C52 C53 D5.D5 C5×D5 C52 C5×D5 C52 D5 C5 C52 C5 C5 C1 # reps 1 1 2 4 4 8 2 2 8 8 1 4 4 16

Matrix representation of C5×D5.D5 in GL4(𝔽41) generated by

 18 0 0 0 0 18 0 0 0 0 18 0 0 0 0 18
,
 10 0 0 0 0 37 0 0 0 0 16 0 2 15 16 18
,
 0 37 0 0 10 0 0 0 39 2 18 33 39 26 25 23
,
 10 0 0 0 0 10 0 0 0 0 37 0 17 24 0 37
,
 0 0 37 0 5 36 37 20 0 10 0 0 30 11 0 5
G:=sub<GL(4,GF(41))| [18,0,0,0,0,18,0,0,0,0,18,0,0,0,0,18],[10,0,0,2,0,37,0,15,0,0,16,16,0,0,0,18],[0,10,39,39,37,0,2,26,0,0,18,25,0,0,33,23],[10,0,0,17,0,10,0,24,0,0,37,0,0,0,0,37],[0,5,0,30,0,36,10,11,37,37,0,0,0,20,0,5] >;

C5×D5.D5 in GAP, Magma, Sage, TeX

C_5\times D_5.D_5
% in TeX

G:=Group("C5xD5.D5");
// GroupNames label

G:=SmallGroup(500,42);
// by ID

G=gap.SmallGroup(500,42);
# by ID

G:=PCGroup([5,-2,-5,-2,-5,-5,50,1603,7504,1014]);
// Polycyclic

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

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