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G = F5×S4order 480 = 25·3·5

Direct product of F5 and S4

direct product, non-abelian, soluble, monomial

Aliases: F5×S4, C5⋊S4⋊C4, C5⋊(C4×S4), (C5×S4)⋊C4, A4⋊F5⋊C2, (A4×F5)⋊C2, (D5×S4).C2, C22⋊(S3×F5), A41(C2×F5), (C22×F5)⋊S3, D5.1(C2×S4), (C22×D5).D6, (D5×A4).C22, (C5×A4)⋊(C2×C4), (C2×C10)⋊(C4×S3), Hol(C2×C10), SmallGroup(480,1189)

Series: Derived Chief Lower central Upper central

C1C22C5×A4 — F5×S4
C1C22C2×C10C5×A4D5×A4A4×F5 — F5×S4
C5×A4 — F5×S4
C1

Generators and relations for F5×S4
 G = < a,b,c,d,e,f | a5=b4=c2=d2=e3=f2=1, bab-1=a3, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, bf=fb, ece-1=fcf=cd=dc, ede-1=c, df=fd, fef=e-1 >

Subgroups: 912 in 112 conjugacy classes, 18 normal (all characteristic)
C1, C2 [×5], C3, C4 [×6], C22, C22 [×6], C5, S3 [×2], C6, C2×C4 [×7], D4 [×4], C23 [×2], D5, D5 [×2], C10 [×2], Dic3, C12, A4, D6, C15, C42, C22⋊C4 [×2], C4⋊C4, C22×C4 [×2], C2×D4, Dic5, C20, F5, F5 [×3], D10 [×5], C2×C10, C2×C10, C4×S3, S4, S4, C2×A4, C5×S3, C3×D5, D15, C4×D4, C4×D5, D20, C5⋊D4 [×2], C5×D4, C2×F5 [×6], C22×D5, C22×D5, A4⋊C4, C4×A4, C2×S4, C3×F5, C3⋊F5, S3×D5, C5×A4, C4×F5, C4⋊F5, C22⋊F5 [×2], D4×D5, C22×F5, C22×F5, C4×S4, S3×F5, C5×S4, C5⋊S4, D5×A4, D4×F5, A4⋊F5, A4×F5, D5×S4, F5×S4
Quotients: C1, C2 [×3], C4 [×2], C22, S3, C2×C4, D6, F5, C4×S3, S4, C2×F5, C2×S4, C4×S4, S3×F5, F5×S4

Character table of F5×S4

 class 12A2B2C2D2E34A4B4C4D4E4F4G4H4I4J5610A10B12A12B1520
 size 13561530855615153030303030440122440403224
ρ11111111111111111111111111    trivial
ρ21111111-1-11-1-1-1-1-1-111111-1-111    linear of order 2
ρ3111-11-11-1-1-1-1-11111-1111-1-1-11-1    linear of order 2
ρ4111-11-1111-111-1-1-1-1-1111-1111-1    linear of order 2
ρ511-11-1-11i-i1-iii-i-ii-11-111i-i11    linear of order 4
ρ611-11-1-11-ii1i-i-iii-i-11-111-ii11    linear of order 4
ρ711-1-1-111-ii-1i-ii-i-ii11-11-1-ii1-1    linear of order 4
ρ811-1-1-111i-i-1-ii-iii-i11-11-1i-i1-1    linear of order 4
ρ9222020-1-2-20-2-2000002-12011-10    orthogonal lifted from D6
ρ10222020-122022000002-120-1-1-10    orthogonal lifted from S3
ρ1122-20-20-1-2i2i02i-2i000002120i-i-10    complex lifted from C4×S3
ρ1222-20-20-12i-2i0-2i2i000002120-ii-10    complex lifted from C4×S3
ρ133-131-110-3-3-1111-11-1-130-11000-1    orthogonal lifted from C2×S4
ρ143-131-11033-1-1-1-11-11-130-11000-1    orthogonal lifted from S4
ρ153-13-1-1-10-3-3111-11-11130-1-10001    orthogonal lifted from C2×S4
ρ163-13-1-1-10331-1-11-11-1130-1-10001    orthogonal lifted from S4
ρ173-1-311-103i-3i-1i-i-i-iii130-11000-1    complex lifted from C4×S4
ρ183-1-311-10-3i3i-1-iiii-i-i130-11000-1    complex lifted from C4×S4
ρ193-1-3-11103i-3i1i-iii-i-i-130-1-10001    complex lifted from C4×S4
ρ203-1-3-1110-3i3i1-ii-i-iii-130-1-10001    complex lifted from C4×S4
ρ21440-400400-40000000-10-1100-11    orthogonal lifted from C2×F5
ρ2244040040040000000-10-1-100-1-1    orthogonal lifted from F5
ρ23880000-40000000000-20-200010    orthogonal lifted from S3×F5
ρ2412-40400000-40000000-301-10001    orthogonal faithful
ρ2512-40-40000040000000-3011000-1    orthogonal faithful

Permutation representations of F5×S4
On 20 points - transitive group 20T121
Generators in S20
(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)(16 19 20 17)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 18)(12 19)(13 20)(14 16)(15 17)
(1 18)(2 19)(3 20)(4 16)(5 17)(6 11)(7 12)(8 13)(9 14)(10 15)
(6 18 11)(7 19 12)(8 20 13)(9 16 14)(10 17 15)
(6 11)(7 12)(8 13)(9 14)(10 15)

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

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

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

G:=TransitiveGroup(20,121);

On 30 points - transitive group 30T110
Generators in S30
(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 9 10 7)(11 14 15 12)(16 19 20 17)(21 24 25 22)(26 29 30 27)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 21)(17 22)(18 23)(19 24)(20 25)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 18 28)(2 19 29)(3 20 30)(4 16 26)(5 17 27)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 16)(7 17)(8 18)(9 19)(10 20)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

G:=TransitiveGroup(30,110);

On 30 points - transitive group 30T114
Generators in S30
(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 30 9 26)(7 27 8 29)(10 28)(16 24 20 22)(17 21 19 25)(18 23)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 21)(17 22)(18 23)(19 24)(20 25)
(6 29)(7 30)(8 26)(9 27)(10 28)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 18 28)(2 19 29)(3 20 30)(4 16 26)(5 17 27)(6 14 24)(7 15 25)(8 11 21)(9 12 22)(10 13 23)
(1 13)(2 14)(3 15)(4 11)(5 12)(6 19)(7 20)(8 16)(9 17)(10 18)(21 26)(22 27)(23 28)(24 29)(25 30)

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

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

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

G:=TransitiveGroup(30,114);

On 30 points - transitive group 30T115
Generators in S30
(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 9 10 7)(11 14 15 12)(16 19 20 17)(21 24 25 22)(26 29 30 27)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 21)(17 22)(18 23)(19 24)(20 25)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 18 28)(2 19 29)(3 20 30)(4 16 26)(5 17 27)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

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

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

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

G:=TransitiveGroup(30,115);

On 30 points - transitive group 30T117
Generators in S30
(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 29 10 27)(7 26 9 30)(8 28)(16 24 20 22)(17 21 19 25)(18 23)
(1 13)(2 14)(3 15)(4 11)(5 12)(16 21)(17 22)(18 23)(19 24)(20 25)
(6 26)(7 27)(8 28)(9 29)(10 30)(16 21)(17 22)(18 23)(19 24)(20 25)
(1 18 28)(2 19 29)(3 20 30)(4 16 26)(5 17 27)(6 11 21)(7 12 22)(8 13 23)(9 14 24)(10 15 25)
(6 21)(7 22)(8 23)(9 24)(10 25)(16 26)(17 27)(18 28)(19 29)(20 30)

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

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

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

G:=TransitiveGroup(30,117);

Matrix representation of F5×S4 in GL7(𝔽61)

1000000
0100000
0010000
00000060
00010060
00001060
00000160
,
50000000
05000000
00500000
00000600
00060000
00000060
00006000
,
06010000
06000000
16000000
0001000
0000100
0000010
0000001
,
01600000
10600000
00600000
0001000
0000100
0000010
0000001
,
0010000
1000000
0100000
0001000
0000100
0000010
0000001
,
0100000
1000000
0010000
00060000
00006000
00000600
00000060

G:=sub<GL(7,GF(61))| [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],[50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,50,0,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,0,60,0,0,0,60,0,0,0,0,0,0,0,0,60,0],[0,0,1,0,0,0,0,60,60,60,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[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],[0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1],[0,1,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60,0,0,0,0,0,0,0,60] >;

F5×S4 in GAP, Magma, Sage, TeX

F_5\times S_4
% in TeX

G:=Group("F5xS4");
// GroupNames label

G:=SmallGroup(480,1189);
// by ID

G=gap.SmallGroup(480,1189);
# by ID

G:=PCGroup([7,-2,-2,-2,-3,-5,-2,2,28,234,1684,858,5052,1286,2953,2232]);
// Polycyclic

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

Export

Character table of F5×S4 in TeX

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