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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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C23 — C5×C2≀C4
 Chief series C1 — C2 — C22 — C23 — C2×D4 — D4×C10 — C5×C23⋊C4 — C5×C2≀C4
 Lower central C1 — C2 — C22 — C23 — C5×C2≀C4
 Upper central C1 — C10 — C2×C10 — D4×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, C4, C22, C22, C5, C8, C2×C4, C2×C4, D4, C23, C23, C10, C10, C22⋊C4, C22⋊C4, M4(2), C2×D4, C2×D4, C24, C20, C2×C10, C2×C10, C23⋊C4, C4.D4, C22≀C2, C40, C2×C20, C2×C20, C5×D4, C22×C10, C22×C10, C2≀C4, C5×C22⋊C4, C5×C22⋊C4, C5×M4(2), D4×C10, D4×C10, C23×C10, C5×C23⋊C4, C5×C4.D4, C5×C22≀C2, C5×C2≀C4
Quotients: C1, C2, C4, C22, C5, C2×C4, D4, C10, C22⋊C4, C20, C2×C10, C23⋊C4, C2×C20, C5×D4, 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 30)(2 26)(3 27)(4 28)(5 29)(6 11)(7 12)(8 13)(9 14)(10 15)(16 36)(17 37)(18 38)(19 39)(20 40)(21 33)(22 34)(23 35)(24 31)(25 32)
(1 35)(2 31)(3 32)(4 33)(5 34)(6 16)(7 17)(8 18)(9 19)(10 20)(11 36)(12 37)(13 38)(14 39)(15 40)(21 28)(22 29)(23 30)(24 26)(25 27)
(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 30)(2 26)(3 27)(4 28)(5 29)(6 36)(7 37)(8 38)(9 39)(10 40)(11 16)(12 17)(13 18)(14 19)(15 20)(21 33)(22 34)(23 35)(24 31)(25 32)
(1 36)(2 37)(3 38)(4 39)(5 40)(6 30)(7 26)(8 27)(9 28)(10 29)(11 35 16 23)(12 31 17 24)(13 32 18 25)(14 33 19 21)(15 34 20 22)

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

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

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

65 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 4A 4B 4C 4D 5A 5B 5C 5D 8A 8B 10A 10B 10C 10D 10E 10F 10G 10H 10I ··· 10X 20A 20B 20C 20D 20E ··· 20P 40A ··· 40H order 1 2 2 2 2 2 2 4 4 4 4 5 5 5 5 8 8 10 10 10 10 10 10 10 10 10 ··· 10 20 20 20 20 20 ··· 20 40 ··· 40 size 1 1 2 4 4 4 4 4 8 8 8 1 1 1 1 8 8 1 1 1 1 2 2 2 2 4 ··· 4 4 4 4 4 8 ··· 8 8 ··· 8

65 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 4 4 4 4 type + + + + + + + + image C1 C2 C2 C2 C4 C4 C5 C10 C10 C10 C20 C20 D4 D4 C5×D4 C5×D4 C23⋊C4 C2≀C4 C5×C23⋊C4 C5×C2≀C4 kernel C5×C2≀C4 C5×C23⋊C4 C5×C4.D4 C5×C22≀C2 C5×C22⋊C4 C23×C10 C2≀C4 C23⋊C4 C4.D4 C22≀C2 C22⋊C4 C24 C2×C20 C22×C10 C2×C4 C23 C10 C5 C2 C1 # reps 1 1 1 1 2 2 4 4 4 4 8 8 1 1 4 4 1 2 4 8

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

 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 39 0 0 0 40 0 0 0 1 0 1 0 1 1 0
,
 1 0 0 0 0 1 0 0 40 0 40 0 40 0 0 40
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 0 40
,
 40 0 39 0 0 0 40 1 0 0 1 0 1 40 1 0
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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