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G = C24⋊D5order 160 = 25·5

The semidirect product of C24 and D5 acting faithfully

non-abelian, soluble, monomial

Aliases: C24⋊D5, C24⋊C5⋊C2, SmallGroup(160,234)

Series: Derived Chief Lower central Upper central

C1C24C24⋊C5 — C24⋊D5
C1C24C24⋊C5 — C24⋊D5
C24⋊C5 — C24⋊D5
C1

Generators and relations for C24⋊D5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f2=1, fdf=ab=ba, ac=ca, ebe-1=fbf=ad=da, eae-1=d, af=fa, bc=cb, bd=db, cd=dc, ece-1=abd, fcf=bcd, ede-1=abcd, fef=e-1 >

5C2
5C2
5C2
20C2
16C5
5C22
5C22
5C22
5C22
5C22
5C22
5C22
10C4
10C4
10C22
10C22
10C4
10C22
16D5
5C23
5C2×C4
5C23
5C23
5C2×C4
5C2×C4
5C23
10D4
10D4
10D4
10D4
10D4
10D4
5C2×D4
5C22⋊C4
5C22⋊C4
5C2×D4
5C22⋊C4
5C2×D4
5C22≀C2

Character table of C24⋊D5

 class 12A2B2C2D4A4B4C5A5B
 size 1555202020203232
ρ11111111111    trivial
ρ21111-1-1-1-111    linear of order 2
ρ322220000-1+5/2-1-5/2    orthogonal lifted from D5
ρ422220000-1-5/2-1+5/2    orthogonal lifted from D5
ρ55-311-111-100    orthogonal faithful
ρ65-3111-1-1100    orthogonal faithful
ρ751-3111-1-100    orthogonal faithful
ρ851-31-1-11100    orthogonal faithful
ρ9511-31-11-100    orthogonal faithful
ρ10511-3-11-1100    orthogonal faithful

Permutation representations of C24⋊D5
On 10 points - transitive group 10T15
Generators in S10
(1 8)(2 9)(4 6)(5 7)
(3 10)(4 6)
(3 10)(5 7)
(1 8)(3 10)(4 6)(5 7)
(1 2 3 4 5)(6 7 8 9 10)
(1 5)(2 4)(6 9)(7 8)

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

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

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

G:=TransitiveGroup(10,15);

On 10 points - transitive group 10T16
Generators in S10
(2 9)(4 6)
(2 9)(3 10)(4 6)(5 7)
(1 8)(2 9)
(1 8)(3 10)
(1 2 3 4 5)(6 7 8 9 10)
(1 7)(2 6)(3 10)(4 9)(5 8)

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

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

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

G:=TransitiveGroup(10,16);

On 16 points: primitive - transitive group 16T415
Generators in S16
(1 7)(2 14)(3 9)(4 5)(6 10)(8 12)(11 16)(13 15)
(1 12)(2 16)(3 13)(4 6)(5 10)(7 8)(9 15)(11 14)
(1 3)(2 10)(4 11)(5 16)(6 14)(7 9)(8 15)(12 13)
(1 11)(2 8)(3 4)(5 9)(6 13)(7 16)(10 15)(12 14)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)
(3 6)(4 5)(8 11)(9 10)(12 16)(13 15)

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

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

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

G:=TransitiveGroup(16,415);

On 20 points - transitive group 20T38
Generators in S20
(2 7)(3 18)(4 14)(8 13)(9 19)(12 17)
(2 7)(3 13)(4 9)(5 15)(8 18)(10 20)(12 17)(14 19)
(1 11)(2 7)(3 18)(4 19)(5 20)(6 16)(8 13)(9 14)(10 15)(12 17)
(1 6)(2 17)(3 13)(7 12)(8 18)(11 16)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 15)(7 14)(8 13)(9 12)(10 11)(16 20)(17 19)

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

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

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

G:=TransitiveGroup(20,38);

On 20 points - transitive group 20T39
Generators in S20
(1 16)(2 7)(4 9)(5 20)(6 11)(10 15)(12 17)(14 19)
(2 12)(3 8)(4 9)(5 15)(7 17)(10 20)(13 18)(14 19)
(1 11)(2 12)(3 18)(5 20)(6 16)(7 17)(8 13)(10 15)
(1 6)(3 8)(4 19)(5 20)(9 14)(10 15)(11 16)(13 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 5)(2 4)(6 10)(7 9)(11 15)(12 14)(16 20)(17 19)

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

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

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

G:=TransitiveGroup(20,39);

On 20 points - transitive group 20T43
Generators in S20
(2 12)(3 13)(4 14)(5 15)(6 19)(7 20)(8 16)(10 18)
(1 11)(2 12)(9 17)(10 18)
(1 11)(3 13)(6 19)(9 17)
(1 11)(2 12)(3 13)(4 14)(6 19)(7 20)(9 17)(10 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 17)(2 16)(3 20)(4 19)(5 18)(6 14)(7 13)(8 12)(9 11)(10 15)

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

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

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

G:=TransitiveGroup(20,43);

On 20 points - transitive group 20T45
Generators in S20
(1 16)(2 11)(4 13)(5 20)(6 14)(7 15)(8 17)(10 19)
(2 8)(3 12)(4 13)(5 6)(9 18)(10 19)(11 17)(14 20)
(1 7)(2 8)(3 18)(5 20)(6 14)(9 12)(11 17)(15 16)
(1 15)(3 12)(4 19)(5 20)(6 14)(7 16)(9 18)(10 13)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 6)(2 10)(3 9)(4 8)(5 7)(11 19)(12 18)(13 17)(14 16)(15 20)

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

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

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

G:=TransitiveGroup(20,45);

C24⋊D5 is a maximal subgroup of   C24⋊F5  F16⋊C2  C24⋊D15
C24⋊D5 is a maximal quotient of   2- 1+4.D5  2- 1+4⋊D5  C25.D5  C24⋊D15

Polynomial with Galois group C24⋊D5 over ℚ
actionf(x)Disc(f)
10T15x10-2x9-14x8+26x7+62x6-110x5-85x4+170x3+8x2-84x+27210·32·672·4014
10T16x10+2x9-103x8+212x7+1931x6-9457x5+14669x4-5930x3-3032x2+503x+83374·672·2812·4015·35812·63432

Matrix representation of C24⋊D5 in GL5(ℤ)

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

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],[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,-1,0,0,0,-1,0,0,0,0,0,0,0,0,-1,0,0,0,-1,0,0,0,-1,0,0] >;

C24⋊D5 in GAP, Magma, Sage, TeX

C_2^4\rtimes D_5
% in TeX

G:=Group("C2^4:D5");
// GroupNames label

G:=SmallGroup(160,234);
// by ID

G=gap.SmallGroup(160,234);
# by ID

G:=PCGroup([6,-2,-5,-2,2,2,2,97,542,278,729,3304,1810,1805,191]);
// Polycyclic

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

Export

Subgroup lattice of C24⋊D5 in TeX
Character table of C24⋊D5 in TeX

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