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## G = C24⋊D5order 160 = 25·5

### The semidirect product of C24 and D5 acting faithfully

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

Series: Derived Chief Lower central Upper central

 Derived series C1 — C24 — C24⋊C5 — C24⋊D5
 Chief series C1 — C24 — C24⋊C5 — C24⋊D5
 Lower central C24⋊C5 — C24⋊D5
 Upper central 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
5C23
5C23
5C23
10D4
10D4
10D4
10D4
10D4
10D4

Character table of C24⋊D5

 class 1 2A 2B 2C 2D 4A 4B 4C 5A 5B size 1 5 5 5 20 20 20 20 32 32 ρ1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 2 2 2 2 0 0 0 0 -1+√5/2 -1-√5/2 orthogonal lifted from D5 ρ4 2 2 2 2 0 0 0 0 -1-√5/2 -1+√5/2 orthogonal lifted from D5 ρ5 5 -3 1 1 -1 1 1 -1 0 0 orthogonal faithful ρ6 5 -3 1 1 1 -1 -1 1 0 0 orthogonal faithful ρ7 5 1 -3 1 1 1 -1 -1 0 0 orthogonal faithful ρ8 5 1 -3 1 -1 -1 1 1 0 0 orthogonal faithful ρ9 5 1 1 -3 1 -1 1 -1 0 0 orthogonal faithful ρ10 5 1 1 -3 -1 1 -1 1 0 0 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 3)(2 10)(4 11)(5 12)(6 15)(7 9)(8 16)(13 14)
(1 11)(2 8)(3 4)(5 9)(6 14)(7 12)(10 16)(13 15)
(1 12)(2 13)(3 5)(4 9)(6 16)(7 11)(8 15)(10 14)
(1 2)(3 10)(4 16)(5 14)(6 9)(7 15)(8 11)(12 13)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)
(2 4)(5 6)(7 9)(10 11)(12 15)(13 14)

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

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

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

G:=TransitiveGroup(16,415);

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

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

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

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

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

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

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

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)

 -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
,
 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

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

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