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G = A4⋊F5order 240 = 24·3·5

The semidirect product of A4 and F5 acting via F5/D5=C2

non-abelian, soluble, monomial

Aliases: A4⋊F5, D5.S4, C5⋊(A4⋊C4), C22⋊(C3⋊F5), (C5×A4)⋊1C4, (C2×C10)⋊Dic3, (D5×A4).1C2, (C22×D5).S3, SmallGroup(240,192)

Series: Derived Chief Lower central Upper central

C1C22C5×A4 — A4⋊F5
C1C22C2×C10C5×A4D5×A4 — A4⋊F5
C5×A4 — A4⋊F5
C1

Generators and relations for A4⋊F5
 G = < a,b,c,d,e | a2=b2=c3=d5=e4=1, cac-1=eae-1=ab=ba, ad=da, cbc-1=a, bd=db, be=eb, cd=dc, ece-1=c-1, ede-1=d3 >

3C2
5C2
15C2
4C3
15C22
15C22
30C4
30C4
20C6
3C10
3D5
4C15
5C23
15C2×C4
15C2×C4
20Dic3
3D10
3D10
6F5
6F5
4C3×D5
15C22⋊C4
5C2×A4
3C2×F5
3C2×F5
4C3⋊F5
5A4⋊C4
3C22⋊F5

Character table of A4⋊F5

 class 12A2B2C34A4B4C4D561015A15B
 size 13515830303030440121616
ρ111111111111111    trivial
ρ211111-1-1-1-111111    linear of order 2
ρ311-1-11-iii-i1-1111    linear of order 4
ρ411-1-11i-i-ii1-1111    linear of order 4
ρ52222-100002-12-1-1    orthogonal lifted from S3
ρ622-2-2-10000212-1-1    symplectic lifted from Dic3, Schur index 2
ρ73-13-101-11-130-100    orthogonal lifted from S4
ρ83-13-10-11-1130-100    orthogonal lifted from S4
ρ93-1-310-i-iii30-100    complex lifted from A4⋊C4
ρ103-1-310ii-i-i30-100    complex lifted from A4⋊C4
ρ11440040000-10-1-1-1    orthogonal lifted from F5
ρ124400-20000-10-11--15/21+-15/2    complex lifted from C3⋊F5
ρ134400-20000-10-11+-15/21--15/2    complex lifted from C3⋊F5
ρ1412-40000000-30100    orthogonal faithful

Permutation representations of A4⋊F5
On 20 points - transitive group 20T61
Generators in S20
(1 20)(2 16)(3 17)(4 18)(5 19)(6 11)(7 12)(8 13)(9 14)(10 15)
(1 10)(2 6)(3 7)(4 8)(5 9)(11 16)(12 17)(13 18)(14 19)(15 20)
(6 11 16)(7 12 17)(8 13 18)(9 14 19)(10 15 20)
(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 7 9 8)(11 17 14 18)(12 19 13 16)(15 20)

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

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

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

G:=TransitiveGroup(20,61);

On 30 points - transitive group 30T53
Generators in S30
(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(2 3 5 4)(6 24 10 22)(7 21 9 25)(8 23)(11 14 15 12)(16 29 20 27)(17 26 19 30)(18 28)

G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,29,20,27)(17,26,19,30)(18,28)>;

G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (2,3,5,4)(6,24,10,22)(7,21,9,25)(8,23)(11,14,15,12)(16,29,20,27)(17,26,19,30)(18,28) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,11),(5,12),(6,26),(7,27),(8,28),(9,29),(10,30)], [(6,26),(7,27),(8,28),(9,29),(10,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,21,11),(7,22,12),(8,23,13),(9,24,14),(10,25,15)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(2,3,5,4),(6,24,10,22),(7,21,9,25),(8,23),(11,14,15,12),(16,29,20,27),(17,26,19,30),(18,28)])

G:=TransitiveGroup(30,53);

On 30 points - transitive group 30T64
Generators in S30
(1 13)(2 14)(3 15)(4 11)(5 12)(6 26)(7 27)(8 28)(9 29)(10 30)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 28 18)(2 29 19)(3 30 20)(4 26 16)(5 27 17)(6 21 11)(7 22 12)(8 23 13)(9 24 14)(10 25 15)
(1 2 3 4 5)(6 7 8 9 10)(11 12 13 14 15)(16 17 18 19 20)(21 22 23 24 25)(26 27 28 29 30)
(1 13)(2 15 5 11)(3 12 4 14)(6 19 10 17)(7 16 9 20)(8 18)(21 29 25 27)(22 26 24 30)(23 28)

G:=sub<Sym(30)| (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,13)(2,15,5,11)(3,12,4,14)(6,19,10,17)(7,16,9,20)(8,18)(21,29,25,27)(22,26,24,30)(23,28)>;

G:=Group( (1,13)(2,14)(3,15)(4,11)(5,12)(6,26)(7,27)(8,28)(9,29)(10,30), (6,26)(7,27)(8,28)(9,29)(10,30)(16,21)(17,22)(18,23)(19,24)(20,25), (1,28,18)(2,29,19)(3,30,20)(4,26,16)(5,27,17)(6,21,11)(7,22,12)(8,23,13)(9,24,14)(10,25,15), (1,2,3,4,5)(6,7,8,9,10)(11,12,13,14,15)(16,17,18,19,20)(21,22,23,24,25)(26,27,28,29,30), (1,13)(2,15,5,11)(3,12,4,14)(6,19,10,17)(7,16,9,20)(8,18)(21,29,25,27)(22,26,24,30)(23,28) );

G=PermutationGroup([(1,13),(2,14),(3,15),(4,11),(5,12),(6,26),(7,27),(8,28),(9,29),(10,30)], [(6,26),(7,27),(8,28),(9,29),(10,30),(16,21),(17,22),(18,23),(19,24),(20,25)], [(1,28,18),(2,29,19),(3,30,20),(4,26,16),(5,27,17),(6,21,11),(7,22,12),(8,23,13),(9,24,14),(10,25,15)], [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15),(16,17,18,19,20),(21,22,23,24,25),(26,27,28,29,30)], [(1,13),(2,15,5,11),(3,12,4,14),(6,19,10,17),(7,16,9,20),(8,18),(21,29,25,27),(22,26,24,30),(23,28)])

G:=TransitiveGroup(30,64);

A4⋊F5 is a maximal subgroup of   F5×S4
A4⋊F5 is a maximal quotient of   C5⋊U2(𝔽3)  D10.S4  Dic5.S4

Matrix representation of A4⋊F5 in GL7(𝔽61)

60000000
60010000
60100000
0001000
0000100
0000010
0000001
,
01600000
10600000
00600000
0001000
0000100
0000010
0000001
,
0100000
0010000
1000000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
00000060
00010060
00001060
00000160
,
01100000
11000000
00110000
0000010
0001000
0000001
0000100

G:=sub<GL(7,GF(61))| [60,60,60,0,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,60,60,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,60,60,60,60],[0,11,0,0,0,0,0,11,0,0,0,0,0,0,0,0,11,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,1,0] >;

A4⋊F5 in GAP, Magma, Sage, TeX

A_4\rtimes F_5
% in TeX

G:=Group("A4:F5");
// GroupNames label

G:=SmallGroup(240,192);
// by ID

G=gap.SmallGroup(240,192);
# by ID

G:=PCGroup([6,-2,-2,-3,-5,-2,2,12,146,867,585,3604,916,2165,1637]);
// Polycyclic

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

Export

Subgroup lattice of A4⋊F5 in TeX
Character table of A4⋊F5 in TeX

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