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), A4⋊1(C2×F5), (C22×F5)⋊S3, D5.1(C2×S4), (D5×A4).C22, (C22×D5).D6, (C5×A4)⋊(C2×C4), (C2×C10)⋊(C4×S3), Hol(C2×C10), SmallGroup(480,1189)
Series: Derived ►Chief ►Lower central ►Upper central
| C5×A4 — F5×S4 |
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
Generators and relations
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 >
►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)(11 13 12 15)(16 17 19 18)
(1 6)(2 7)(3 8)(4 9)(5 10)(11 17)(12 18)(13 19)(14 20)(15 16)
(1 14)(2 15)(3 11)(4 12)(5 13)(6 20)(7 16)(8 17)(9 18)(10 19)
(6 14 20)(7 15 16)(8 11 17)(9 12 18)(10 13 19)
(6 20)(7 16)(8 17)(9 18)(10 19)
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)(11,13,12,15)(16,17,19,18), (1,6)(2,7)(3,8)(4,9)(5,10)(11,17)(12,18)(13,19)(14,20)(15,16), (1,14)(2,15)(3,11)(4,12)(5,13)(6,20)(7,16)(8,17)(9,18)(10,19), (6,14,20)(7,15,16)(8,11,17)(9,12,18)(10,13,19), (6,20)(7,16)(8,17)(9,18)(10,19)>;
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)(11,13,12,15)(16,17,19,18), (1,6)(2,7)(3,8)(4,9)(5,10)(11,17)(12,18)(13,19)(14,20)(15,16), (1,14)(2,15)(3,11)(4,12)(5,13)(6,20)(7,16)(8,17)(9,18)(10,19), (6,14,20)(7,15,16)(8,11,17)(9,12,18)(10,13,19), (6,20)(7,16)(8,17)(9,18)(10,19) );
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),(11,13,12,15),(16,17,19,18)], [(1,6),(2,7),(3,8),(4,9),(5,10),(11,17),(12,18),(13,19),(14,20),(15,16)], [(1,14),(2,15),(3,11),(4,12),(5,13),(6,20),(7,16),(8,17),(9,18),(10,19)], [(6,14,20),(7,15,16),(8,11,17),(9,12,18),(10,13,19)], [(6,20),(7,16),(8,17),(9,18),(10,19)])
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 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)
(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), (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), (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), (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), (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)], [(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)], [(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(30,114);
►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 7 9 8)(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 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)
(6 24)(7 25)(8 21)(9 22)(10 23)(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)(6,7,9,8)(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), (6,29)(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), (6,24)(7,25)(8,21)(9,22)(10,23)(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)(6,7,9,8)(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), (6,29)(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), (6,24)(7,25)(8,21)(9,22)(10,23)(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),(6,7,9,8),(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)], [(6,29),(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)], [(6,24),(7,25),(8,21),(9,22),(10,23),(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 ►G ⊆ GL7(𝔽61)
| 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 | 0 | 0 | 60 |
| 0 | 0 | 0 | 1 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 1 | 0 | 60 |
| 0 | 0 | 0 | 0 | 0 | 1 | 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 | 0 | 60 | 0 |
| 0 | 0 | 0 | 60 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 60 |
| 0 | 0 | 0 | 0 | 60 | 0 | 0 |
| 0 | 60 | 1 | 0 | 0 | 0 | 0 |
| 0 | 60 | 0 | 0 | 0 | 0 | 0 |
| 1 | 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 |
| 0 | 1 | 60 | 0 | 0 | 0 | 0 |
| 1 | 0 | 60 | 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 |
| 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 |
| 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 |
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] >;
Character table of F5×S4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 5 | 6 | 10A | 10B | 12A | 12B | 15 | 20 | |
| size | 1 | 3 | 5 | 6 | 15 | 30 | 8 | 5 | 5 | 6 | 15 | 15 | 30 | 30 | 30 | 30 | 30 | 4 | 40 | 12 | 24 | 40 | 40 | 32 | 24 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 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 | 1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | linear of order 2 |
| ρ5 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | i | -i | 1 | -i | i | i | -i | -i | i | -1 | 1 | -1 | 1 | 1 | i | -i | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -i | i | 1 | i | -i | -i | i | i | -i | -1 | 1 | -1 | 1 | 1 | -i | i | 1 | 1 | linear of order 4 |
| ρ7 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -i | i | -1 | i | -i | i | -i | -i | i | 1 | 1 | -1 | 1 | -1 | -i | i | 1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | i | -i | -1 | -i | i | -i | i | i | -i | 1 | 1 | -1 | 1 | -1 | i | -i | 1 | -1 | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 0 | 2 | 0 | -1 | -2 | -2 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | 0 | 1 | 1 | -1 | 0 | orthogonal lifted from D6 |
| ρ10 | 2 | 2 | 2 | 0 | 2 | 0 | -1 | 2 | 2 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 2 | -1 | 2 | 0 | -1 | -1 | -1 | 0 | orthogonal lifted from S3 |
| ρ11 | 2 | 2 | -2 | 0 | -2 | 0 | -1 | 2i | 2i | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 2 | 0 | i | -i | -1 | 0 | complex lifted from C4×S3 |
| ρ12 | 2 | 2 | -2 | 0 | -2 | 0 | -1 | 2i | 2i | 0 | 2i | 2i | 0 | 0 | 0 | 0 | 0 | 2 | 1 | 2 | 0 | -i | i | -1 | 0 | complex lifted from C4×S3 |
| ρ13 | 3 | -1 | 3 | 1 | -1 | 1 | 0 | -3 | -3 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 3 | 0 | -1 | 1 | 0 | 0 | 0 | -1 | orthogonal lifted from C2×S4 |
| ρ14 | 3 | -1 | 3 | 1 | -1 | 1 | 0 | 3 | 3 | -1 | -1 | -1 | -1 | 1 | -1 | 1 | -1 | 3 | 0 | -1 | 1 | 0 | 0 | 0 | -1 | orthogonal lifted from S4 |
| ρ15 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | -3 | -3 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 3 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | orthogonal lifted from C2×S4 |
| ρ16 | 3 | -1 | 3 | -1 | -1 | -1 | 0 | 3 | 3 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | 1 | 3 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | orthogonal lifted from S4 |
| ρ17 | 3 | -1 | -3 | 1 | 1 | -1 | 0 | 3i | 3i | -1 | i | -i | -i | -i | i | i | 1 | 3 | 0 | -1 | 1 | 0 | 0 | 0 | -1 | complex lifted from C4×S4 |
| ρ18 | 3 | -1 | -3 | 1 | 1 | -1 | 0 | 3i | 3i | -1 | -i | i | i | i | -i | -i | 1 | 3 | 0 | -1 | 1 | 0 | 0 | 0 | -1 | complex lifted from C4×S4 |
| ρ19 | 3 | -1 | -3 | -1 | 1 | 1 | 0 | 3i | 3i | 1 | i | -i | i | i | -i | -i | -1 | 3 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | complex lifted from C4×S4 |
| ρ20 | 3 | -1 | -3 | -1 | 1 | 1 | 0 | 3i | 3i | 1 | -i | i | -i | -i | i | i | -1 | 3 | 0 | -1 | -1 | 0 | 0 | 0 | 1 | complex lifted from C4×S4 |
| ρ21 | 4 | 4 | 0 | -4 | 0 | 0 | 4 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | -1 | 1 | 0 | 0 | -1 | 1 | orthogonal lifted from C2×F5 |
| ρ22 | 4 | 4 | 0 | 4 | 0 | 0 | 4 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | -1 | -1 | 0 | 0 | -1 | -1 | orthogonal lifted from F5 |
| ρ23 | 8 | 8 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 0 | -2 | 0 | 0 | 0 | 1 | 0 | orthogonal lifted from S3×F5 |
| ρ24 | 12 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 1 | -1 | 0 | 0 | 0 | 1 | orthogonal faithful |
| ρ25 | 12 | -4 | 0 | -4 | 0 | 0 | 0 | 0 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3 | 0 | 1 | 1 | 0 | 0 | 0 | -1 | orthogonal faithful |
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
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ℤ
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