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## G = C2×C24⋊C5order 160 = 25·5

### Direct product of C2 and C24⋊C5

Aliases: C2×C24⋊C5, C2C5, AΣL1(𝔽32), C25⋊C5, C24⋊C10, SmallGroup(160,235)

Series: Derived Chief Lower central Upper central

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

Generators and relations for C2×C24⋊C5
G = < a,b,c,d,e,f | a2=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: 408 in 82 conjugacy classes, 6 normal (all characteristic)
C1, C2, C2 [×6], C22 [×31], C5, C23 [×31], C10, C24, C24 [×6], C25, C24⋊C5, C2×C24⋊C5
Quotients: C1, C2, C5, C10, C24⋊C5, C2×C24⋊C5

Character table of C2×C24⋊C5

 class 1 2A 2B 2C 2D 2E 2F 2G 5A 5B 5C 5D 10A 10B 10C 10D size 1 1 5 5 5 5 5 5 16 16 16 16 16 16 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 -1 1 1 1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ3 1 1 1 1 1 1 1 1 ζ54 ζ53 ζ52 ζ5 ζ5 ζ53 ζ52 ζ54 linear of order 5 ρ4 1 1 1 1 1 1 1 1 ζ5 ζ52 ζ53 ζ54 ζ54 ζ52 ζ53 ζ5 linear of order 5 ρ5 1 -1 1 1 1 -1 -1 -1 ζ5 ζ52 ζ53 ζ54 -ζ54 -ζ52 -ζ53 -ζ5 linear of order 10 ρ6 1 -1 1 1 1 -1 -1 -1 ζ52 ζ54 ζ5 ζ53 -ζ53 -ζ54 -ζ5 -ζ52 linear of order 10 ρ7 1 1 1 1 1 1 1 1 ζ53 ζ5 ζ54 ζ52 ζ52 ζ5 ζ54 ζ53 linear of order 5 ρ8 1 1 1 1 1 1 1 1 ζ52 ζ54 ζ5 ζ53 ζ53 ζ54 ζ5 ζ52 linear of order 5 ρ9 1 -1 1 1 1 -1 -1 -1 ζ53 ζ5 ζ54 ζ52 -ζ52 -ζ5 -ζ54 -ζ53 linear of order 10 ρ10 1 -1 1 1 1 -1 -1 -1 ζ54 ζ53 ζ52 ζ5 -ζ5 -ζ53 -ζ52 -ζ54 linear of order 10 ρ11 5 5 1 -3 1 1 1 -3 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ12 5 -5 1 -3 1 -1 -1 3 0 0 0 0 0 0 0 0 orthogonal faithful ρ13 5 -5 1 1 -3 3 -1 -1 0 0 0 0 0 0 0 0 orthogonal faithful ρ14 5 -5 -3 1 1 -1 3 -1 0 0 0 0 0 0 0 0 orthogonal faithful ρ15 5 5 -3 1 1 1 -3 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5 ρ16 5 5 1 1 -3 -3 1 1 0 0 0 0 0 0 0 0 orthogonal lifted from C24⋊C5

Permutation representations of C2×C24⋊C5
On 10 points - transitive group 10T14
Generators in S10
(1 8)(2 9)(3 10)(4 6)(5 7)
(1 8)(3 10)
(1 8)(2 9)(4 6)(5 7)
(3 10)(4 6)
(2 9)(4 6)
(1 2 3 4 5)(6 7 8 9 10)

G:=sub<Sym(10)| (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(3,10), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (2,9)(4,6), (1,2,3,4,5)(6,7,8,9,10)>;

G:=Group( (1,8)(2,9)(3,10)(4,6)(5,7), (1,8)(3,10), (1,8)(2,9)(4,6)(5,7), (3,10)(4,6), (2,9)(4,6), (1,2,3,4,5)(6,7,8,9,10) );

G=PermutationGroup([(1,8),(2,9),(3,10),(4,6),(5,7)], [(1,8),(3,10)], [(1,8),(2,9),(4,6),(5,7)], [(3,10),(4,6)], [(2,9),(4,6)], [(1,2,3,4,5),(6,7,8,9,10)])

G:=TransitiveGroup(10,14);

On 20 points - transitive group 20T40
Generators in S20
(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 12)(8 11)(9 18)(15 16)
(1 7)(2 11)(4 10)(5 14)(6 20)(8 17)(13 19)(15 16)
(1 16)(2 17)(3 12)(4 10)(5 20)(6 14)(7 15)(8 11)(9 18)(13 19)
(2 8)(3 18)(4 13)(9 12)(10 19)(11 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)| (1,7)(2,8)(3,9)(4,10)(5,6)(11,17)(12,18)(13,19)(14,20)(15,16), (1,7)(2,17)(3,12)(8,11)(9,18)(15,16), (1,7)(2,11)(4,10)(5,14)(6,20)(8,17)(13,19)(15,16), (1,16)(2,17)(3,12)(4,10)(5,20)(6,14)(7,15)(8,11)(9,18)(13,19), (2,8)(3,18)(4,13)(9,12)(10,19)(11,17), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)>;

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

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

G:=TransitiveGroup(20,40);

On 20 points - transitive group 20T41
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 17)(3 8)(6 16)(7 12)(13 18)
(1 11)(2 7)(4 14)(5 10)(6 16)(9 19)(12 17)(15 20)
(1 16)(2 17)(3 8)(4 14)(5 20)(6 11)(7 12)(9 19)(10 15)(13 18)
(2 12)(3 18)(4 9)(7 17)(8 13)(14 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)| (1,16)(2,17)(3,18)(4,19)(5,20)(6,11)(7,12)(8,13)(9,14)(10,15), (1,11)(2,17)(3,8)(6,16)(7,12)(13,18), (1,11)(2,7)(4,14)(5,10)(6,16)(9,19)(12,17)(15,20), (1,16)(2,17)(3,8)(4,14)(5,20)(6,11)(7,12)(9,19)(10,15)(13,18), (2,12)(3,18)(4,9)(7,17)(8,13)(14,19), (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)(2,17)(3,8)(6,16)(7,12)(13,18), (1,11)(2,7)(4,14)(5,10)(6,16)(9,19)(12,17)(15,20), (1,16)(2,17)(3,8)(4,14)(5,20)(6,11)(7,12)(9,19)(10,15)(13,18), (2,12)(3,18)(4,9)(7,17)(8,13)(14,19), (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),(2,17),(3,8),(6,16),(7,12),(13,18)], [(1,11),(2,7),(4,14),(5,10),(6,16),(9,19),(12,17),(15,20)], [(1,16),(2,17),(3,8),(4,14),(5,20),(6,11),(7,12),(9,19),(10,15),(13,18)], [(2,12),(3,18),(4,9),(7,17),(8,13),(14,19)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20)])

G:=TransitiveGroup(20,41);

On 20 points - transitive group 20T44
Generators in S20
(1 17)(2 18)(3 19)(4 20)(5 16)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 13)(3 15)(8 17)(10 19)
(1 13)(2 14)(4 11)(5 12)(6 20)(7 16)(8 17)(9 18)
(3 15)(4 11)(6 20)(10 19)
(2 14)(4 11)(6 20)(9 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,44);

On 20 points - transitive group 20T46
Generators in S20
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 20)
(1 11)(3 13)(4 19)(5 20)(6 16)(8 18)(9 14)(10 15)
(1 11)(2 7)(4 9)(5 15)(6 16)(10 20)(12 17)(14 19)
(2 17)(3 8)(4 9)(5 20)(7 12)(10 15)(13 18)(14 19)
(1 16)(2 12)(4 14)(5 20)(6 11)(7 17)(9 19)(10 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)

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

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

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

G:=TransitiveGroup(20,46);

C2×C24⋊C5 is a maximal subgroup of   C25.D5
C2×C24⋊C5 is a maximal quotient of   2- 1+4.C10

Polynomial with Galois group C2×C24⋊C5 over ℚ
actionf(x)Disc(f)
10T14x10-12x8+51x6-96x4+80x2-23210·118·23

Matrix representation of C2×C24⋊C5 in GL5(ℤ)

 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1
,
 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1
,
 -1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 -1
,
 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 1 0 0 0 0

G:=sub<GL(5,Integers())| [-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,-1],[-1,0,0,0,0,0,-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1],[-1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,1],[-1,0,0,0,0,0,-1,0,0,0,0,0,1,0,0,0,0,0,-1,0,0,0,0,0,-1],[0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0] >;

C2×C24⋊C5 in GAP, Magma, Sage, TeX

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

G:=Group("C2xC2^4:C5");
// GroupNames label

G:=SmallGroup(160,235);
// by ID

G=gap.SmallGroup(160,235);
# by ID

G:=PCGroup([6,-2,-5,-2,2,2,2,728,1089,1660,2711]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=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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