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## G = A4×F5order 240 = 24·3·5

### Direct product of A4 and F5

Aliases: A4×F5, C5⋊(C4×A4), (C2×C10)⋊C12, D5.(C2×A4), (C5×A4)⋊2C4, C22⋊(C3×F5), (C22×F5)⋊C3, (D5×A4).2C2, (C22×D5).C6, SmallGroup(240,193)

Series: Derived Chief Lower central Upper central

 Derived series C1 — C2×C10 — A4×F5
 Chief series C1 — C5 — C2×C10 — C22×D5 — D5×A4 — A4×F5
 Lower central C2×C10 — A4×F5
 Upper central C1

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

3C2
5C2
15C2
4C3
5C4
15C4
15C22
15C22
20C6
3C10
3D5
4C15
5C23
15C2×C4
15C2×C4
20C12
3D10
3F5
3D10

Character table of A4×F5

 class 1 2A 2B 2C 3A 3B 4A 4B 4C 4D 5 6A 6B 10 12A 12B 12C 12D 15A 15B size 1 3 5 15 4 4 5 5 15 15 4 20 20 12 20 20 20 20 16 16 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ3 1 1 1 1 ζ32 ζ3 -1 -1 -1 -1 1 ζ32 ζ3 1 ζ65 ζ6 ζ65 ζ6 ζ3 ζ32 linear of order 6 ρ4 1 1 1 1 ζ32 ζ3 1 1 1 1 1 ζ32 ζ3 1 ζ3 ζ32 ζ3 ζ32 ζ3 ζ32 linear of order 3 ρ5 1 1 1 1 ζ3 ζ32 1 1 1 1 1 ζ3 ζ32 1 ζ32 ζ3 ζ32 ζ3 ζ32 ζ3 linear of order 3 ρ6 1 1 1 1 ζ3 ζ32 -1 -1 -1 -1 1 ζ3 ζ32 1 ζ6 ζ65 ζ6 ζ65 ζ32 ζ3 linear of order 6 ρ7 1 1 -1 -1 1 1 i -i -i i 1 -1 -1 1 -i -i i i 1 1 linear of order 4 ρ8 1 1 -1 -1 1 1 -i i i -i 1 -1 -1 1 i i -i -i 1 1 linear of order 4 ρ9 1 1 -1 -1 ζ3 ζ32 i -i -i i 1 ζ65 ζ6 1 ζ43ζ32 ζ43ζ3 ζ4ζ32 ζ4ζ3 ζ32 ζ3 linear of order 12 ρ10 1 1 -1 -1 ζ32 ζ3 -i i i -i 1 ζ6 ζ65 1 ζ4ζ3 ζ4ζ32 ζ43ζ3 ζ43ζ32 ζ3 ζ32 linear of order 12 ρ11 1 1 -1 -1 ζ3 ζ32 -i i i -i 1 ζ65 ζ6 1 ζ4ζ32 ζ4ζ3 ζ43ζ32 ζ43ζ3 ζ32 ζ3 linear of order 12 ρ12 1 1 -1 -1 ζ32 ζ3 i -i -i i 1 ζ6 ζ65 1 ζ43ζ3 ζ43ζ32 ζ4ζ3 ζ4ζ32 ζ3 ζ32 linear of order 12 ρ13 3 -1 3 -1 0 0 -3 -3 1 1 3 0 0 -1 0 0 0 0 0 0 orthogonal lifted from C2×A4 ρ14 3 -1 3 -1 0 0 3 3 -1 -1 3 0 0 -1 0 0 0 0 0 0 orthogonal lifted from A4 ρ15 3 -1 -3 1 0 0 -3i 3i -i i 3 0 0 -1 0 0 0 0 0 0 complex lifted from C4×A4 ρ16 3 -1 -3 1 0 0 3i -3i i -i 3 0 0 -1 0 0 0 0 0 0 complex lifted from C4×A4 ρ17 4 4 0 0 4 4 0 0 0 0 -1 0 0 -1 0 0 0 0 -1 -1 orthogonal lifted from F5 ρ18 4 4 0 0 -2+2√-3 -2-2√-3 0 0 0 0 -1 0 0 -1 0 0 0 0 ζ6 ζ65 complex lifted from C3×F5 ρ19 4 4 0 0 -2-2√-3 -2+2√-3 0 0 0 0 -1 0 0 -1 0 0 0 0 ζ65 ζ6 complex lifted from C3×F5 ρ20 12 -4 0 0 0 0 0 0 0 0 -3 0 0 1 0 0 0 0 0 0 orthogonal faithful

Permutation representations of A4×F5
On 20 points - transitive group 20T68
Generators in S20
(1 6)(2 7)(3 8)(4 9)(5 10)(11 16)(12 17)(13 18)(14 19)(15 20)
(1 11)(2 12)(3 13)(4 14)(5 15)(6 16)(7 17)(8 18)(9 19)(10 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)(7 8 10 9)(12 13 15 14)(17 18 20 19)

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

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

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

G:=TransitiveGroup(20,68);

On 30 points - transitive group 30T56
Generators in S30
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)
(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 6)(2 8 5 9)(3 10 4 7)(11 16)(12 18 15 19)(13 20 14 17)(21 26)(22 28 25 29)(23 30 24 27)

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

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

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

G:=TransitiveGroup(30,56);

On 30 points - transitive group 30T57
Generators in S30
(1 6)(2 7)(3 8)(4 9)(5 10)(21 26)(22 27)(23 28)(24 29)(25 30)
(11 16)(12 17)(13 18)(14 19)(15 20)(21 26)(22 27)(23 28)(24 29)(25 30)
(1 21 11)(2 22 12)(3 23 13)(4 24 14)(5 25 15)(6 26 16)(7 27 17)(8 28 18)(9 29 19)(10 30 20)
(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)(7 8 10 9)(12 13 15 14)(17 18 20 19)(22 23 25 24)(27 28 30 29)

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

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

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

G:=TransitiveGroup(30,57);

A4×F5 is a maximal quotient of   SL2(𝔽3).F5

Matrix representation of A4×F5 in GL7(𝔽61)

 1 5 0 0 0 0 0 0 60 0 0 0 0 0 48 59 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
,
 1 0 5 0 0 0 0 48 60 59 0 0 0 0 0 0 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
,
 1 0 0 0 0 0 0 48 60 60 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 60 60 60 60 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0
,
 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 60 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 60 60 60 60

G:=sub<GL(7,GF(61))| [1,0,48,0,0,0,0,5,60,59,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,0,60,0,0,0,0,0,5,59,60,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,48,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,1,0,0,0,0,0,60,0,1,0,0,0,0,60,0,0,1,0,0,0,60,0,0,0],[60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,1,0,0,60,0,0,0,0,0,1,60,0,0,0,0,0,0,60,0,0,0,0,1,0,60] >;

A4×F5 in GAP, Magma, Sage, TeX

A_4\times F_5
% in TeX

G:=Group("A4xF5");
// GroupNames label

G:=SmallGroup(240,193);
// by ID

G=gap.SmallGroup(240,193);
# by ID

G:=PCGroup([6,-2,-3,-2,-2,2,-5,36,441,190,3461,1169]);
// Polycyclic

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

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