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G = C4×C24⋊C5  order 320 = 26·5

Direct product of C4 and C24⋊C5

direct product, metabelian, soluble, monomial, A-group

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, C4, C4, C22, C5, C2×C4, C23, C10, C22×C4, C24, C24, C20, C23×C4, 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 11 17)(2 9 12 18)(3 10 13 19)(4 6 14 20)(5 7 15 16)

(1 11)(2 12)(4 14)(5 15)(6 20)(7 16)(8 17)(9 18)

(1 11)(2 12)(8 17)(9 18)

(1 11)(3 13)(8 17)(10 19)

(1 11)(2 12)(3 13)(5 15)(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)| (1,8,11,17)(2,9,12,18)(3,10,13,19)(4,6,14,20)(5,7,15,16), (1,11)(2,12)(4,14)(5,15)(6,20)(7,16)(8,17)(9,18), (1,11)(2,12)(8,17)(9,18), (1,11)(3,13)(8,17)(10,19), (1,11)(2,12)(3,13)(5,15)(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( (1,8,11,17)(2,9,12,18)(3,10,13,19)(4,6,14,20)(5,7,15,16), (1,11)(2,12)(4,14)(5,15)(6,20)(7,16)(8,17)(9,18), (1,11)(2,12)(8,17)(9,18), (1,11)(3,13)(8,17)(10,19), (1,11)(2,12)(3,13)(5,15)(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([[(1,8,11,17),(2,9,12,18),(3,10,13,19),(4,6,14,20),(5,7,15,16)], [(1,11),(2,12),(4,14),(5,15),(6,20),(7,16),(8,17),(9,18)], [(1,11),(2,12),(8,17),(9,18)], [(1,11),(3,13),(8,17),(10,19)], [(1,11),(2,12),(3,13),(5,15),(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 2A2B···2G4A4B4C···4H5A5B5C5D10A10B10C10D20A···20H
order122···2444···455551010101020···20
size115···5115···5161616161616161616···16

32 irreducible representations

dim111111555
type++++
imageC1C2C4C5C10C20C24⋊C5C2×C24⋊C5C4×C24⋊C5
kernelC4×C24⋊C5C2×C24⋊C5C24⋊C5C24×C4C25C24C4C2C1
# reps112448336

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

320000
032000
003200
000320
000032
,
400000
251000
310100
000400
230001
,
400000
040000
004000
40010
000040
,
10000
01000
1004000
3700400
00001
,
10000
01000
1004000
00010
1800040
,
1639000
025100
031010
04001
023000

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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