Copied to
clipboard

G = C2×C24⋊C5order 160 = 25·5

Direct product of C2 and C24⋊C5

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

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

Series: Derived Chief Lower central Upper central

C1C24 — C2×C24⋊C5
C1C24C24⋊C5 — C2×C24⋊C5
C24 — C2×C24⋊C5
C1C2

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 12A2B2C2D2E2F2G5A5B5C5D10A10B10C10D
 size 115555551616161616161616
ρ11111111111111111    trivial
ρ21-1111-1-1-11111-1-1-1-1    linear of order 2
ρ311111111ζ54ζ53ζ52ζ5ζ5ζ53ζ52ζ54    linear of order 5
ρ411111111ζ5ζ52ζ53ζ54ζ54ζ52ζ53ζ5    linear of order 5
ρ51-1111-1-1-1ζ5ζ52ζ53ζ545452535    linear of order 10
ρ61-1111-1-1-1ζ52ζ54ζ5ζ535354552    linear of order 10
ρ711111111ζ53ζ5ζ54ζ52ζ52ζ5ζ54ζ53    linear of order 5
ρ811111111ζ52ζ54ζ5ζ53ζ53ζ54ζ5ζ52    linear of order 5
ρ91-1111-1-1-1ζ53ζ5ζ54ζ525255453    linear of order 10
ρ101-1111-1-1-1ζ54ζ53ζ52ζ55535254    linear of order 10
ρ11551-3111-300000000    orthogonal lifted from C24⋊C5
ρ125-51-31-1-1300000000    orthogonal faithful
ρ135-511-33-1-100000000    orthogonal faithful
ρ145-5-311-13-100000000    orthogonal faithful
ρ1555-3111-3100000000    orthogonal lifted from C24⋊C5
ρ165511-3-31100000000    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)

-10000
0-1000
00-100
000-10
0000-1
,
-10000
0-1000
00-100
00010
0000-1
,
-10000
01000
00100
00010
0000-1
,
-10000
01000
00100
000-10
00001
,
-10000
0-1000
00100
000-10
0000-1
,
01000
00100
00010
00001
10000

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

Export

Character table of C2×C24⋊C5 in TeX

׿
×
𝔽