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G = C22×C24⋊C5order 320 = 26·5

Direct product of C22 and C24⋊C5

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

Aliases: C22×C24⋊C5, C26⋊C5, C25⋊C10, C24⋊(C2×C10), SmallGroup(320,1637)

Series: Derived Chief Lower central Upper central

C1C24 — C22×C24⋊C5
C1C24C24⋊C5C2×C24⋊C5 — C22×C24⋊C5
C24 — C22×C24⋊C5
C1C22

Generators and relations for C22×C24⋊C5
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=e2=f2=g5=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, bc=cb, bd=db, be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, gcg-1=cde, de=ed, df=fd, gdg-1=def, geg-1=ef=fe, gfg-1=c >

Subgroups: 2910 in 583 conjugacy classes, 15 normal (6 characteristic)
C1, C2 [×3], C2 [×12], C22, C22 [×130], C5, C23 [×279], C10 [×3], C24, C24 [×130], C2×C10, C25 [×3], C25 [×12], C26, C24⋊C5, C2×C24⋊C5 [×3], C22×C24⋊C5
Quotients: C1, C2 [×3], C22, C5, C10 [×3], C2×C10, C24⋊C5, C2×C24⋊C5 [×3], C22×C24⋊C5

Permutation representations of C22×C24⋊C5
On 20 points - transitive group 20T72
Generators in S20
(1 16)(2 17)(3 18)(4 19)(5 20)(6 14)(7 15)(8 11)(9 12)(10 13)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 7)(2 17)(3 18)(4 10)(8 11)(9 12)(13 19)(15 16)
(2 11)(3 9)(4 10)(5 14)(6 20)(8 17)(12 18)(13 19)
(1 15)(2 11)(3 18)(5 20)(6 14)(7 16)(8 17)(9 12)
(2 8)(3 18)(4 19)(5 6)(9 12)(10 13)(11 17)(14 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,72);

On 20 points - transitive group 20T74
Generators in S20
(1 15)(2 11)(3 12)(4 13)(5 14)(6 20)(7 16)(8 17)(9 18)(10 19)
(1 7)(2 8)(3 9)(4 10)(5 6)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 15)(4 10)(5 20)(6 14)(7 16)(13 19)
(2 8)(3 12)(4 10)(5 14)(6 20)(9 18)(11 17)(13 19)
(1 15)(2 8)(3 18)(4 19)(5 20)(6 14)(7 16)(9 12)(10 13)(11 17)
(1 16)(2 11)(5 6)(7 15)(8 17)(14 20)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,74);

On 20 points - transitive group 20T86
Generators in S20
(1 13)(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)(10 19)
(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)
(2 14)(3 15)(4 11)(5 12)(6 20)(7 16)(9 18)(10 19)
(4 11)(5 12)(6 20)(7 16)
(1 13)(4 11)(6 20)(8 17)
(1 13)(3 15)(4 11)(5 12)(6 20)(7 16)(8 17)(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)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(9,18)(10,19), (1,17)(2,18)(3,19)(4,20)(5,16)(6,11)(7,12)(8,13)(9,14)(10,15), (2,14)(3,15)(4,11)(5,12)(6,20)(7,16)(9,18)(10,19), (4,11)(5,12)(6,20)(7,16), (1,13)(4,11)(6,20)(8,17), (1,13)(3,15)(4,11)(5,12)(6,20)(7,16)(8,17)(10,19), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

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

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

G:=TransitiveGroup(20,86);

32 conjugacy classes

class 1 2A2B2C2D···2O5A5B5C5D10A···10L
order12222···2555510···10
size11115···51616161616···16

32 irreducible representations

dim111155
type++++
imageC1C2C5C10C24⋊C5C2×C24⋊C5
kernelC22×C24⋊C5C2×C24⋊C5C26C25C22C2
# reps1341239

Matrix representation of C22×C24⋊C5 in GL6(𝔽11)

1000000
0100000
0010000
0001000
0000100
0000010
,
100000
0100000
0010000
0001000
0000100
0000010
,
100000
010000
0010000
0001000
000010
000001
,
100000
010000
001000
0001000
000010
0000010
,
100000
010000
0010000
0001000
0000100
0000010
,
100000
0100000
0010000
000100
000010
000001
,
900000
001000
000100
000010
000001
010000

G:=sub<GL(6,GF(11))| [10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,10],[1,0,0,0,0,0,0,10,0,0,0,0,0,0,10,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[9,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0] >;

C22×C24⋊C5 in GAP, Magma, Sage, TeX

C_2^2\times C_2^4\rtimes C_5
% in TeX

G:=Group("C2^2xC2^4:C5");
// GroupNames label

G:=SmallGroup(320,1637);
// by ID

G=gap.SmallGroup(320,1637);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,1137,1593,2329,3695]);
// Polycyclic

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

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