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## G = C22×C24⋊C5order 320 = 26·5

### Direct product of C22 and C24⋊C5

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C22×C24⋊C5
 Chief series C1 — C24 — C24⋊C5 — C2×C24⋊C5 — C22×C24⋊C5
 Lower central C24 — C22×C24⋊C5
 Upper central C1 — C22

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, C2, C22, C22, C5, C23, C10, C24, C24, C2×C10, C25, C25, C26, C24⋊C5, C2×C24⋊C5, C22×C24⋊C5
Quotients: C1, C2, C22, C5, C10, C2×C10, C24⋊C5, C2×C24⋊C5, 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 11)(7 12)(8 13)(9 14)(10 15)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 11)(2 17)(3 18)(4 14)(6 16)(7 12)(8 13)(9 19)
(2 7)(3 13)(4 14)(5 10)(8 18)(9 19)(12 17)(15 20)
(1 6)(2 7)(3 18)(5 20)(8 13)(10 15)(11 16)(12 17)
(2 12)(3 18)(4 19)(5 15)(7 17)(8 13)(9 14)(10 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)(6,11)(7,12)(8,13)(9,14)(10,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,6)(2,7)(3,18)(5,20)(8,13)(10,15)(11,16)(12,17), (2,12)(3,18)(4,19)(5,15)(7,17)(8,13)(9,14)(10,20), (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)(6,11)(7,12)(8,13)(9,14)(10,15), (1,11)(2,12)(3,13)(4,14)(5,15)(6,16)(7,17)(8,18)(9,19)(10,20), (1,11)(2,17)(3,18)(4,14)(6,16)(7,12)(8,13)(9,19), (2,7)(3,13)(4,14)(5,10)(8,18)(9,19)(12,17)(15,20), (1,6)(2,7)(3,18)(5,20)(8,13)(10,15)(11,16)(12,17), (2,12)(3,18)(4,19)(5,15)(7,17)(8,13)(9,14)(10,20), (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),(6,11),(7,12),(8,13),(9,14),(10,15)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,11),(2,17),(3,18),(4,14),(6,16),(7,12),(8,13),(9,19)], [(2,7),(3,13),(4,14),(5,10),(8,18),(9,19),(12,17),(15,20)], [(1,6),(2,7),(3,18),(5,20),(8,13),(10,15),(11,16),(12,17)], [(2,12),(3,18),(4,19),(5,15),(7,17),(8,13),(9,14),(10,20)], [(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 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 6)(4 14)(5 20)(9 19)(10 15)(11 16)
(2 12)(3 8)(4 14)(5 10)(7 17)(9 19)(13 18)(15 20)
(1 6)(2 12)(3 18)(4 19)(5 20)(7 17)(8 13)(9 14)(10 15)(11 16)
(1 16)(2 7)(5 15)(6 11)(10 20)(12 17)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,9),(5,10),(11,16),(12,17),(13,18),(14,19),(15,20)], [(1,11),(2,12),(3,13),(4,14),(5,15),(6,16),(7,17),(8,18),(9,19),(10,20)], [(1,6),(4,14),(5,20),(9,19),(10,15),(11,16)], [(2,12),(3,8),(4,14),(5,10),(7,17),(9,19),(13,18),(15,20)], [(1,6),(2,12),(3,18),(4,19),(5,20),(7,17),(8,13),(9,14),(10,15),(11,16)], [(1,16),(2,7),(5,15),(6,11),(10,20),(12,17)], [(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 2A 2B 2C 2D ··· 2O 5A 5B 5C 5D 10A ··· 10L order 1 2 2 2 2 ··· 2 5 5 5 5 10 ··· 10 size 1 1 1 1 5 ··· 5 16 16 16 16 16 ··· 16

32 irreducible representations

 dim 1 1 1 1 5 5 type + + + + image C1 C2 C5 C10 C24⋊C5 C2×C24⋊C5 kernel C22×C24⋊C5 C2×C24⋊C5 C26 C25 C22 C2 # reps 1 3 4 12 3 9

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

 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 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 0 0 0 0

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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