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G = C5×C4.Dic5order 400 = 24·52

Direct product of C5 and C4.Dic5

direct product, metacyclic, supersoluble, monomial

Aliases: C5×C4.Dic5, C20.4C20, C20.67D10, C102.11C4, C20.12Dic5, C5217M4(2), C52C85C10, C4.(C5×Dic5), (C2×C20).7C10, (C10×C20).7C2, (C2×C10).7C20, (C5×C20).14C4, (C2×C20).18D5, C4.15(D5×C10), C54(C5×M4(2)), C20.16(C2×C10), C10.13(C2×C20), C22.(C5×Dic5), C2.3(C10×Dic5), (C2×C10).3Dic5, (C5×C20).45C22, C10.25(C2×Dic5), (C5×C52C8)⋊12C2, (C2×C4).2(C5×D5), (C5×C10).61(C2×C4), SmallGroup(400,82)

Series: Derived Chief Lower central Upper central

C1C10 — C5×C4.Dic5
C1C5C10C20C5×C20C5×C52C8 — C5×C4.Dic5
C5C10 — C5×C4.Dic5
C1C20C2×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 [×2], C22, C5 [×2], C5 [×2], C8 [×2], C2×C4, C10 [×2], C10 [×8], M4(2), C20 [×4], C20 [×4], C2×C10 [×2], C2×C10 [×2], C52, C52C8 [×2], C40 [×2], C2×C20 [×2], C2×C20 [×2], C5×C10, C5×C10, C4.Dic5, C5×M4(2), C5×C20 [×2], C102, C5×C52C8 [×2], C10×C20, C5×C4.Dic5
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D5, C10 [×3], M4(2), Dic5 [×2], C20 [×2], D10, C2×C10, C2×Dic5, C2×C20, C5×D5, C4.Dic5, C5×M4(2), C5×Dic5 [×2], 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 36 6 21 11 26 16 31)(2 25 7 30 12 35 17 40)(3 34 8 39 13 24 18 29)(4 23 9 28 14 33 19 38)(5 32 10 37 15 22 20 27)

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

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

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

130 conjugacy classes

class 1 2A2B4A4B4C5A5B5C5D5E···5N8A8B8C8D10A10B10C10D10E···10AL20A···20H20I···20AZ40A···40P
order12244455555···588881010101010···1020···2020···2040···40
size11211211112···21010101011112···21···12···210···10

130 irreducible representations

dim1111111111222222222222
type++++-+-
imageC1C2C2C4C4C5C10C10C20C20D5M4(2)Dic5D10Dic5C5×D5C4.Dic5C5×M4(2)C5×Dic5D5×C10C5×Dic5C5×C4.Dic5
kernelC5×C4.Dic5C5×C52C8C10×C20C5×C20C102C4.Dic5C52C8C2×C20C20C2×C10C2×C20C52C20C20C2×C10C2×C4C5C5C4C4C22C1
# reps12122484882222288888832

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

180
018
,
92
032
,
392
021
,
2418
3017
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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