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## G = C4×C24⋊C5order 320 = 26·5

### Direct product of C4 and C24⋊C5

Aliases: C4×C24⋊C5, C24⋊C20, C25.C10, (C24×C4)⋊C5, C2.1(C2×C24⋊C5), (C2×C24⋊C5).2C2, SmallGroup(320,1584)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C4×C24⋊C5
 Chief series C1 — C24 — C25 — C2×C24⋊C5 — C4×C24⋊C5
 Lower central C24 — C4×C24⋊C5
 Upper central C1 — C4

Generators and relations for C4×C24⋊C5
G = < a,b,c,d,e,f | a4=b2=c2=d2=e2=f5=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcd, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, fef-1=b >

Subgroups: 732 in 147 conjugacy classes, 9 normal (all characteristic)
C1, C2, C2 [×6], C4, C4 [×3], C22 [×31], C5, C2×C4 [×24], C23 [×31], C10, C22×C4 [×28], C24, C24 [×6], C20, C23×C4 [×6], C25, C24×C4, C24⋊C5, C2×C24⋊C5, C4×C24⋊C5
Quotients: C1, C2, C4, C5, C10, C20, C24⋊C5, C2×C24⋊C5, C4×C24⋊C5

Permutation representations of C4×C24⋊C5
On 20 points - transitive group 20T75
Generators in S20
(1 8 13 17)(2 9 14 18)(3 10 15 19)(4 6 11 20)(5 7 12 16)
(1 13)(2 14)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)
(1 13)(2 14)(8 17)(9 18)
(1 13)(3 15)(8 17)(10 19)
(1 13)(2 14)(3 15)(5 12)(7 16)(8 17)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

G:=sub<Sym(20)| (1,8,13,17)(2,9,14,18)(3,10,15,19)(4,6,11,20)(5,7,12,16), (1,13)(2,14)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18), (1,13)(2,14)(8,17)(9,18), (1,13)(3,15)(8,17)(10,19), (1,13)(2,14)(3,15)(5,12)(7,16)(8,17)(9,18)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

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

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

G:=TransitiveGroup(20,75);

32 conjugacy classes

 class 1 2A 2B ··· 2G 4A 4B 4C ··· 4H 5A 5B 5C 5D 10A 10B 10C 10D 20A ··· 20H order 1 2 2 ··· 2 4 4 4 ··· 4 5 5 5 5 10 10 10 10 20 ··· 20 size 1 1 5 ··· 5 1 1 5 ··· 5 16 16 16 16 16 16 16 16 16 ··· 16

32 irreducible representations

 dim 1 1 1 1 1 1 5 5 5 type + + + + image C1 C2 C4 C5 C10 C20 C24⋊C5 C2×C24⋊C5 C4×C24⋊C5 kernel C4×C24⋊C5 C2×C24⋊C5 C24⋊C5 C24×C4 C25 C24 C4 C2 C1 # reps 1 1 2 4 4 8 3 3 6

Matrix representation of C4×C24⋊C5 in GL5(𝔽41)

 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32 0 0 0 0 0 32
,
 40 0 0 0 0 25 1 0 0 0 31 0 1 0 0 0 0 0 40 0 23 0 0 0 1
,
 40 0 0 0 0 0 40 0 0 0 0 0 40 0 0 4 0 0 1 0 0 0 0 0 40
,
 1 0 0 0 0 0 1 0 0 0 10 0 40 0 0 37 0 0 40 0 0 0 0 0 1
,
 1 0 0 0 0 0 1 0 0 0 10 0 40 0 0 0 0 0 1 0 18 0 0 0 40
,
 16 39 0 0 0 0 25 1 0 0 0 31 0 1 0 0 4 0 0 1 0 23 0 0 0

G:=sub<GL(5,GF(41))| [32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32,0,0,0,0,0,32],[40,25,31,0,23,0,1,0,0,0,0,0,1,0,0,0,0,0,40,0,0,0,0,0,1],[40,0,0,4,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40],[1,0,10,37,0,0,1,0,0,0,0,0,40,0,0,0,0,0,40,0,0,0,0,0,1],[1,0,10,0,18,0,1,0,0,0,0,0,40,0,0,0,0,0,1,0,0,0,0,0,40],[16,0,0,0,0,39,25,31,4,23,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C4×C24⋊C5 in GAP, Magma, Sage, TeX

C_4\times C_2^4\rtimes C_5
% in TeX

G:=Group("C4xC2^4:C5");
// GroupNames label

G:=SmallGroup(320,1584);
// by ID

G=gap.SmallGroup(320,1584);
# by ID

G:=PCGroup([7,-2,-5,-2,-2,2,2,2,70,2250,3161,4632,7363]);
// Polycyclic

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

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