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## G = C5×C4.Dic5order 400 = 24·52

### Direct product of C5 and C4.Dic5

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — C5×C4.Dic5
 Chief series C1 — C5 — C10 — C20 — C5×C20 — C5×C5⋊2C8 — C5×C4.Dic5
 Lower central C5 — C10 — C5×C4.Dic5
 Upper central C1 — C20 — C2×C20

Generators and relations for C5×C4.Dic5
G = < a,b,c,d | a5=b4=1, c10=b2, d2=c5, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c9 >

Subgroups: 100 in 56 conjugacy classes, 30 normal (26 characteristic)
C1, C2, C2, C4, C22, C5, C5, C8, C2×C4, C10, C10, M4(2), C20, C20, C2×C10, C2×C10, C52, C52C8, C40, C2×C20, C2×C20, C5×C10, C5×C10, C4.Dic5, C5×M4(2), C5×C20, C102, C5×C52C8, C10×C20, C5×C4.Dic5
Quotients: C1, C2, C4, C22, C5, C2×C4, D5, C10, M4(2), Dic5, C20, D10, C2×C10, C2×Dic5, C2×C20, C5×D5, C4.Dic5, C5×M4(2), C5×Dic5, D5×C10, C10×Dic5, C5×C4.Dic5

Smallest permutation representation of C5×C4.Dic5
On 40 points
Generators in S40
(1 17 13 9 5)(2 18 14 10 6)(3 19 15 11 7)(4 20 16 12 8)(21 25 29 33 37)(22 26 30 34 38)(23 27 31 35 39)(24 28 32 36 40)
(1 6 11 16)(2 7 12 17)(3 8 13 18)(4 9 14 19)(5 10 15 20)(21 36 31 26)(22 37 32 27)(23 38 33 28)(24 39 34 29)(25 40 35 30)
(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 30 6 35 11 40 16 25)(2 39 7 24 12 29 17 34)(3 28 8 33 13 38 18 23)(4 37 9 22 14 27 19 32)(5 26 10 31 15 36 20 21)

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

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

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

130 conjugacy classes

 class 1 2A 2B 4A 4B 4C 5A 5B 5C 5D 5E ··· 5N 8A 8B 8C 8D 10A 10B 10C 10D 10E ··· 10AL 20A ··· 20H 20I ··· 20AZ 40A ··· 40P order 1 2 2 4 4 4 5 5 5 5 5 ··· 5 8 8 8 8 10 10 10 10 10 ··· 10 20 ··· 20 20 ··· 20 40 ··· 40 size 1 1 2 1 1 2 1 1 1 1 2 ··· 2 10 10 10 10 1 1 1 1 2 ··· 2 1 ··· 1 2 ··· 2 10 ··· 10

130 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 type + + + + - + - image C1 C2 C2 C4 C4 C5 C10 C10 C20 C20 D5 M4(2) Dic5 D10 Dic5 C5×D5 C4.Dic5 C5×M4(2) C5×Dic5 D5×C10 C5×Dic5 C5×C4.Dic5 kernel C5×C4.Dic5 C5×C5⋊2C8 C10×C20 C5×C20 C102 C4.Dic5 C5⋊2C8 C2×C20 C20 C2×C10 C2×C20 C52 C20 C20 C2×C10 C2×C4 C5 C5 C4 C4 C22 C1 # reps 1 2 1 2 2 4 8 4 8 8 2 2 2 2 2 8 8 8 8 8 8 32

Matrix representation of C5×C4.Dic5 in GL2(𝔽41) generated by

 18 0 0 18
,
 9 2 0 32
,
 39 2 0 21
,
 24 18 30 17
G:=sub<GL(2,GF(41))| [18,0,0,18],[9,0,2,32],[39,0,2,21],[24,30,18,17] >;

C5×C4.Dic5 in GAP, Magma, Sage, TeX

C_5\times C_4.{\rm Dic}_5
% in TeX

G:=Group("C5xC4.Dic5");
// GroupNames label

G:=SmallGroup(400,82);
// by ID

G=gap.SmallGroup(400,82);
# by ID

G:=PCGroup([6,-2,-2,-5,-2,-2,-5,120,505,69,11525]);
// Polycyclic

G:=Group<a,b,c,d|a^5=b^4=1,c^10=b^2,d^2=c^5,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^9>;
// generators/relations

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