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G = C24⋊F5order 320 = 26·5

The semidirect product of C24 and F5 acting faithfully

non-abelian, soluble, monomial

Aliases: C24⋊F5, C24⋊C5⋊C4, C24⋊D5.C2, SmallGroup(320,1635)

Series: Derived Chief Lower central Upper central

C1C24C24⋊C5 — C24⋊F5
C1C24C24⋊C5C24⋊D5 — C24⋊F5
C24⋊C5 — C24⋊F5
C1

Generators and relations for C24⋊F5
 G = < a,b,c,d,e,f | a2=b2=c2=d2=e5=f4=1, fdf-1=ab=ba, ac=ca, ebe-1=fcf-1=ad=da, eae-1=faf-1=d, bc=cb, bd=db, fbf-1=acd, cd=dc, ece-1=abd, ede-1=abcd, fef-1=e3 >

5C2
10C2
20C2
16C5
5C22
10C22
10C22
10C22
10C4
10C22
20C22
20C4
40C4
16D5
5C23
5C23
5C2×C4
10C2×C4
10C23
20C2×C4
20C8
20D4
20D4
20D4
16F5
5C22⋊C4
5C2×D4
10C2×D4
10C22⋊C4
10C22⋊C4
10M4(2)
5C4.D4
5C22≀C2
5C23⋊C4
5C2≀C4

Character table of C24⋊F5

 class 12A2B2C4A4B4C4D58A8B
 size 15102020404040644040
ρ111111111111    trivial
ρ2111111-1-11-1-1    linear of order 2
ρ3111-1-1-1-ii1i-i    linear of order 4
ρ4111-1-1-1i-i1-ii    linear of order 4
ρ544400000-100    orthogonal lifted from F5
ρ65-3111-1110-1-1    orthogonal faithful
ρ75-3111-1-1-1011    orthogonal faithful
ρ85-31-1-11-ii0-ii    complex faithful
ρ95-31-1-11i-i0i-i    complex faithful
ρ10102-22-2000000    orthogonal faithful
ρ11102-2-22000000    orthogonal faithful

Permutation representations of C24⋊F5
On 10 points - transitive group 10T24
Generators in S10
(1 10)(2 6)(4 8)(5 9)
(3 7)(4 8)
(3 7)(5 9)
(1 10)(3 7)(4 8)(5 9)
(1 2 3 4 5)(6 7 8 9 10)
(2 3 5 4)(6 7 9 8)

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

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

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

G:=TransitiveGroup(10,24);

On 10 points - transitive group 10T25
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 8)(2 10 5 6)(3 7 4 9)

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), (1,8)(2,10,5,6)(3,7,4,9)>;

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), (1,8)(2,10,5,6)(3,7,4,9) );

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

G:=TransitiveGroup(10,25);

On 16 points: primitive - transitive group 16T711
Generators in S16
(1 9)(2 12)(3 11)(4 5)(6 7)(8 14)(10 15)(13 16)
(1 15)(2 14)(3 16)(4 6)(5 7)(8 12)(9 10)(11 13)
(1 3)(2 7)(4 8)(5 14)(6 12)(9 11)(10 13)(15 16)
(1 8)(2 10)(3 4)(5 11)(6 16)(7 13)(9 14)(12 15)
(2 3 4 5 6)(7 8 9 10 11)(12 13 14 15 16)
(2 14 3 16)(4 13 6 12)(5 15)(8 9 11 10)

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

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

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

G:=TransitiveGroup(16,711);

On 20 points - transitive group 20T77
Generators in S20
(1 10)(2 16)(4 18)(5 9)(6 11)(8 13)(14 19)(15 20)
(2 11)(3 17)(4 18)(5 14)(6 16)(7 12)(8 13)(9 19)
(1 15)(2 11)(3 7)(5 9)(6 16)(10 20)(12 17)(14 19)
(1 20)(3 17)(4 8)(5 9)(7 12)(10 15)(13 18)(14 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 20 15 10)(2 17 14 8)(3 19 13 6)(4 16 12 9)(5 18 11 7)

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

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

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

G:=TransitiveGroup(20,77);

On 20 points - transitive group 20T78
Generators in S20
(2 11)(4 13)(9 18)(10 19)
(2 11)(3 12)(4 13)(5 14)(6 20)(9 18)
(1 15)(2 11)(6 20)(7 16)(9 18)(10 19)
(1 15)(3 12)(8 17)(9 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10 15 19)(2 7 14 17)(3 9 13 20)(4 6 12 18)(5 8 11 16)

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

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

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

G:=TransitiveGroup(20,78);

On 20 points - transitive group 20T79
Generators in S20
(1 15)(2 11)(4 13)(5 14)(6 20)(8 17)(9 18)(10 19)
(3 12)(4 13)(7 16)(8 17)
(3 12)(5 14)(7 16)(9 18)
(1 15)(3 12)(4 13)(5 14)(7 16)(8 17)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 16 5 17)(3 18 4 20)(6 12 9 13)(7 14 8 11)(10 15)

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

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

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

G:=TransitiveGroup(20,79);

On 20 points - transitive group 20T80
Generators in S20
(1 10)(2 16)(4 18)(5 9)(6 11)(8 13)(14 19)(15 20)
(2 11)(3 17)(4 18)(5 14)(6 16)(7 12)(8 13)(9 19)
(1 15)(2 11)(3 7)(5 9)(6 16)(10 20)(12 17)(14 19)
(1 20)(3 17)(4 8)(5 9)(7 12)(10 15)(13 18)(14 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(2 3 5 4)(6 17 9 18)(7 19 8 16)(10 20)(11 12 14 13)

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

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

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

G:=TransitiveGroup(20,80);

On 20 points - transitive group 20T83
Generators in S20
(2 11)(4 13)(9 18)(10 19)
(2 11)(3 12)(4 13)(5 14)(6 20)(9 18)
(1 15)(2 11)(6 20)(7 16)(9 18)(10 19)
(1 15)(3 12)(8 17)(9 18)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 19)(2 16 5 17)(3 18 4 20)(6 12 9 13)(7 14 8 11)(10 15)

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

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

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

G:=TransitiveGroup(20,83);

On 20 points - transitive group 20T88
Generators in S20
(1 15)(2 11)(4 13)(5 14)(6 20)(8 17)(9 18)(10 19)
(3 12)(4 13)(7 16)(8 17)
(3 12)(5 14)(7 16)(9 18)
(1 15)(3 12)(4 13)(5 14)(7 16)(8 17)(9 18)(10 19)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)
(1 10 15 19)(2 7 14 17)(3 9 13 20)(4 6 12 18)(5 8 11 16)

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

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

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

G:=TransitiveGroup(20,88);

Polynomial with Galois group C24⋊F5 over ℚ
actionf(x)Disc(f)
10T24x10-13x8+56x6-90x4+40x2-4228·536
10T25x10-25x8+160x6-400x4+395x2-125218·513·434

Matrix representation of C24⋊F5 in GL5(ℤ)

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

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

C24⋊F5 in GAP, Magma, Sage, TeX

C_2^4\rtimes F_5
% in TeX

G:=Group("C2^4:F5");
// GroupNames label

G:=SmallGroup(320,1635);
// by ID

G=gap.SmallGroup(320,1635);
# by ID

G:=PCGroup([7,-2,-2,-5,-2,2,2,2,14,170,177,2803,850,1137,9104,1593,5045,4632,2329,986,2463,3695]);
// Polycyclic

G:=Group<a,b,c,d,e,f|a^2=b^2=c^2=d^2=e^5=f^4=1,f*d*f^-1=a*b=b*a,a*c=c*a,e*b*e^-1=f*c*f^-1=a*d=d*a,e*a*e^-1=f*a*f^-1=d,b*c=c*b,b*d=d*b,f*b*f^-1=a*c*d,c*d=d*c,e*c*e^-1=a*b*d,e*d*e^-1=a*b*c*d,f*e*f^-1=e^3>;
// generators/relations

Export

Subgroup lattice of C24⋊F5 in TeX
Character table of C24⋊F5 in TeX

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