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G = C5×C2≀C4order 320 = 26·5

Direct product of C5 and C2≀C4

direct product, metabelian, nilpotent (class 4), monomial, 2-elementary

Aliases: C5×C2≀C4, C242C20, C23⋊C41C10, C22⋊C41C20, (C23×C10)⋊1C4, (C2×C20).17D4, C23.1(C5×D4), C4.D45C10, C23.1(C2×C20), (C22×C10).1D4, C22≀C2.1C10, C10.53(C23⋊C4), (D4×C10).174C22, (C2×C4).1(C5×D4), (C5×C23⋊C4)⋊7C2, (C5×C22⋊C4)⋊9C4, C2.6(C5×C23⋊C4), (C2×D4).1(C2×C10), (C5×C22≀C2).3C2, (C5×C4.D4)⋊12C2, (C22×C10).33(C2×C4), C22.10(C5×C22⋊C4), (C2×C10).137(C22⋊C4), SmallGroup(320,156)

Series: Derived Chief Lower central Upper central

C1C23 — C5×C2≀C4
C1C2C22C23C2×D4D4×C10C5×C23⋊C4 — C5×C2≀C4
C1C2C22C23 — C5×C2≀C4
C1C10C2×C10D4×C10 — C5×C2≀C4

Generators and relations for C5×C2≀C4
 G = < a,b,c,d,e,f | a5=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >

Subgroups: 258 in 94 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2 [×5], C4 [×3], C22, C22 [×12], C5, C8, C2×C4, C2×C4 [×2], D4 [×3], C23 [×2], C23 [×4], C10, C10 [×5], C22⋊C4, C22⋊C4 [×2], M4(2), C2×D4, C2×D4, C24, C20 [×3], C2×C10, C2×C10 [×12], C23⋊C4, C4.D4, C22≀C2, C40, C2×C20, C2×C20 [×2], C5×D4 [×3], C22×C10 [×2], C22×C10 [×4], C2≀C4, C5×C22⋊C4, C5×C22⋊C4 [×2], C5×M4(2), D4×C10, D4×C10, C23×C10, C5×C23⋊C4, C5×C4.D4, C5×C22≀C2, C5×C2≀C4
Quotients: C1, C2 [×3], C4 [×2], C22, C5, C2×C4, D4 [×2], C10 [×3], C22⋊C4, C20 [×2], C2×C10, C23⋊C4, C2×C20, C5×D4 [×2], C2≀C4, C5×C22⋊C4, C5×C23⋊C4, C5×C2≀C4

Smallest permutation representation of C5×C2≀C4
On 40 points
Generators in S40
(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 21)(2 22)(3 23)(4 24)(5 25)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(21 31)(22 32)(23 33)(24 34)(25 35)
(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 21)(2 22)(3 23)(4 24)(5 25)(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)(26 31)(27 32)(28 33)(29 34)(30 35)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 21)(7 22)(8 23)(9 24)(10 25)(11 26 16 31)(12 27 17 32)(13 28 18 33)(14 29 19 34)(15 30 20 35)

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

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

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

65 conjugacy classes

class 1 2A2B2C2D2E2F4A4B4C4D5A5B5C5D8A8B10A10B10C10D10E10F10G10H10I···10X20A20B20C20D20E···20P40A···40H
order12222224444555588101010101010101010···102020202020···2040···40
size11244444888111188111122224···444448···88···8

65 irreducible representations

dim11111111111122224444
type++++++++
imageC1C2C2C2C4C4C5C10C10C10C20C20D4D4C5×D4C5×D4C23⋊C4C2≀C4C5×C23⋊C4C5×C2≀C4
kernelC5×C2≀C4C5×C23⋊C4C5×C4.D4C5×C22≀C2C5×C22⋊C4C23×C10C2≀C4C23⋊C4C4.D4C22≀C2C22⋊C4C24C2×C20C22×C10C2×C4C23C10C5C2C1
# reps11112244448811441248

Matrix representation of C5×C2≀C4 in GL4(𝔽41) generated by

37000
03700
00370
00037
,
40000
04000
00040
00400
,
13900
04000
0101
0110
,
1000
0100
400400
400040
,
40000
04000
00400
00040
,
400390
00401
0010
14010
G:=sub<GL(4,GF(41))| [37,0,0,0,0,37,0,0,0,0,37,0,0,0,0,37],[40,0,0,0,0,40,0,0,0,0,0,40,0,0,40,0],[1,0,0,0,39,40,1,1,0,0,0,1,0,0,1,0],[1,0,40,40,0,1,0,0,0,0,40,0,0,0,0,40],[40,0,0,0,0,40,0,0,0,0,40,0,0,0,0,40],[40,0,0,1,0,0,0,40,39,40,1,1,0,1,0,0] >;

C5×C2≀C4 in GAP, Magma, Sage, TeX

C_5\times C_2\wr C_4
% in TeX

G:=Group("C5xC2wrC4");
// GroupNames label

G:=SmallGroup(320,156);
// by ID

G=gap.SmallGroup(320,156);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,-2,-2,-2,280,309,2803,2111,375,10085]);
// Polycyclic

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

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