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G = A4×F5order 240 = 24·3·5

Direct product of A4 and F5

direct product, metabelian, soluble, monomial, A-group

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

C1C2×C10 — A4×F5
C1C5C2×C10C22×D5D5×A4 — A4×F5
C2×C10 — A4×F5
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
4C3×D5
5C22×C4
5C2×A4
3C2×F5
3C2×F5
4C3×F5
5C4×A4

Character table of A4×F5

 class 12A2B2C3A3B4A4B4C4D56A6B1012A12B12C12D15A15B
 size 13515445515154202012202020201616
ρ111111111111111111111    trivial
ρ2111111-1-1-1-11111-1-1-1-111    linear of order 2
ρ31111ζ32ζ3-1-1-1-11ζ32ζ31ζ65ζ6ζ65ζ6ζ3ζ32    linear of order 6
ρ41111ζ32ζ311111ζ32ζ31ζ3ζ32ζ3ζ32ζ3ζ32    linear of order 3
ρ51111ζ3ζ3211111ζ3ζ321ζ32ζ3ζ32ζ3ζ32ζ3    linear of order 3
ρ61111ζ3ζ32-1-1-1-11ζ3ζ321ζ6ζ65ζ6ζ65ζ32ζ3    linear of order 6
ρ711-1-111i-i-ii1-1-11-i-iii11    linear of order 4
ρ811-1-111-iii-i1-1-11ii-i-i11    linear of order 4
ρ911-1-1ζ3ζ32i-i-ii1ζ65ζ61ζ43ζ32ζ43ζ3ζ4ζ32ζ4ζ3ζ32ζ3    linear of order 12
ρ1011-1-1ζ32ζ3-iii-i1ζ6ζ651ζ4ζ3ζ4ζ32ζ43ζ3ζ43ζ32ζ3ζ32    linear of order 12
ρ1111-1-1ζ3ζ32-iii-i1ζ65ζ61ζ4ζ32ζ4ζ3ζ43ζ32ζ43ζ3ζ32ζ3    linear of order 12
ρ1211-1-1ζ32ζ3i-i-ii1ζ6ζ651ζ43ζ3ζ43ζ32ζ4ζ3ζ4ζ32ζ3ζ32    linear of order 12
ρ133-13-100-3-311300-1000000    orthogonal lifted from C2×A4
ρ143-13-10033-1-1300-1000000    orthogonal lifted from A4
ρ153-1-3100-3i3i-ii300-1000000    complex lifted from C4×A4
ρ163-1-31003i-3ii-i300-1000000    complex lifted from C4×A4
ρ174400440000-100-10000-1-1    orthogonal lifted from F5
ρ184400-2+2-3-2-2-30000-100-10000ζ6ζ65    complex lifted from C3×F5
ρ194400-2-2-3-2+2-30000-100-10000ζ65ζ6    complex lifted from C3×F5
ρ2012-400000000-3001000000    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)

1500000
06000000
4859600000
0001000
0000100
0000010
0000001
,
1050000
4860590000
00600000
0001000
0000100
0000010
0000001
,
1000000
4860600000
0100000
0001000
0000100
0000010
0000001
,
1000000
0100000
0010000
00060606060
0001000
0000100
0000010
,
60000000
06000000
00600000
0001000
0000001
0000100
00060606060

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

Export

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

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