Aliases: C24.2A4, C23.2SL2(𝔽3), C23.Q8⋊C3, C23.17(C2×A4), C2.C42⋊1C6, C23.3A4⋊2C2, C2.2(C42⋊C6), C22.3(C2×SL2(𝔽3)), SmallGroup(192,197)
Series: Derived ►Chief ►Lower central ►Upper central
| C2.C42 — C24.2A4 |
Generators and relations for C24.2A4
G = < a,b,c,d,e,f,g | a2=b2=c2=d2=g3=1, e2=gbg-1=bcd, f2=gcg-1=b, ab=ba, ac=ca, ad=da, eae-1=abd, faf-1=acd, ag=ga, bc=cb, bd=db, be=eb, bf=fb, cd=dc, ce=ec, cf=fc, fef-1=de=ed, df=fd, dg=gd, geg-1=bef, gfg-1=cde >
3C2
3C2
4C2
4C2
16C3
3C22
3C22
4C22
6C22
6C4
6C4
6C22
12C4
12C22
16C6
16C6
16C6
6C23
6C23
4A4
16C2×C6
Character table of C24.2A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 3 | 3 | 4 | 4 | 16 | 16 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | |
| ρ1 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | -1 | ζ32 | ζ3 | 1 | 1 | -1 | -1 | ζ65 | ζ3 | ζ6 | ζ6 | ζ65 | ζ32 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | -1 | -1 | ζ3 | ζ32 | 1 | 1 | -1 | -1 | ζ6 | ζ32 | ζ65 | ζ65 | ζ6 | ζ3 | linear of order 6 |
| ρ7 | 2 | -2 | 2 | -2 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | -1 | 1 | 1 | -1 | 1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ8 | 2 | -2 | 2 | -2 | -2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 1 | -1 | 1 | symplectic lifted from SL2(𝔽3), Schur index 2 |
| ρ9 | 2 | -2 | 2 | -2 | -2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ3 | ζ3 | ζ6 | ζ32 | ζ65 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | -2 | 2 | -2 | -2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ32 | ζ32 | ζ65 | ζ3 | ζ6 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | -2 | 2 | -2 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | ζ65 | ζ3 | ζ32 | ζ6 | ζ3 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | ζ6 | ζ32 | ζ3 | ζ65 | ζ32 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | 3 | 3 | 3 | -3 | -3 | 0 | 0 | -1 | -1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ16 | 6 | -6 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal faithful |
| ρ17 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | 2i | -2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
| ρ18 | 6 | 6 | -2 | -2 | 0 | 0 | 0 | 0 | -2i | 2i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C6 |
►On 12 points - transitive group 12T91
Generators in S12
(1 4)(2 3)(5 6)(7 8)(9 10)(11 12)
(5 7)(6 8)
(1 2)(3 4)
(1 2)(3 4)(5 7)(6 8)(9 11)(10 12)
(3 4)(5 6)(7 8)(9 10 11 12)
(1 3)(2 4)(5 6 7 8)(9 11)
(1 10 7)(2 12 5)(3 11 6)(4 9 8)
G:=sub<Sym(12)| (1,4)(2,3)(5,6)(7,8)(9,10)(11,12), (5,7)(6,8), (1,2)(3,4), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(9,11), (1,10,7)(2,12,5)(3,11,6)(4,9,8)>;
G:=Group( (1,4)(2,3)(5,6)(7,8)(9,10)(11,12), (5,7)(6,8), (1,2)(3,4), (1,2)(3,4)(5,7)(6,8)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(9,11), (1,10,7)(2,12,5)(3,11,6)(4,9,8) );
G=PermutationGroup([[(1,4),(2,3),(5,6),(7,8),(9,10),(11,12)], [(5,7),(6,8)], [(1,2),(3,4)], [(1,2),(3,4),(5,7),(6,8),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,3),(2,4),(5,6,7,8),(9,11)], [(1,10,7),(2,12,5),(3,11,6),(4,9,8)]])
G:=TransitiveGroup(12,91);
►On 12 points - transitive group 12T93
Generators in S12
(3 8)(5 6)(10 12)
(1 2)(5 6)
(3 8)(4 7)
(1 2)(3 8)(4 7)(5 6)(9 11)(10 12)
(3 4)(5 6)(7 8)(9 10 11 12)
(1 5 2 6)(4 7)(9 10)(11 12)
(1 7 9)(2 4 11)(3 12 5)(6 8 10)
G:=sub<Sym(12)| (3,8)(5,6)(10,12), (1,2)(5,6), (3,8)(4,7), (1,2)(3,8)(4,7)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(4,7)(9,10)(11,12), (1,7,9)(2,4,11)(3,12,5)(6,8,10)>;
G:=Group( (3,8)(5,6)(10,12), (1,2)(5,6), (3,8)(4,7), (1,2)(3,8)(4,7)(5,6)(9,11)(10,12), (3,4)(5,6)(7,8)(9,10,11,12), (1,5,2,6)(4,7)(9,10)(11,12), (1,7,9)(2,4,11)(3,12,5)(6,8,10) );
G=PermutationGroup([[(3,8),(5,6),(10,12)], [(1,2),(5,6)], [(3,8),(4,7)], [(1,2),(3,8),(4,7),(5,6),(9,11),(10,12)], [(3,4),(5,6),(7,8),(9,10,11,12)], [(1,5,2,6),(4,7),(9,10),(11,12)], [(1,7,9),(2,4,11),(3,12,5),(6,8,10)]])
G:=TransitiveGroup(12,93);
►On 24 points - transitive group 24T460
Generators in S24
(1 7)(2 14)(3 8)(4 13)(5 15)(6 16)(9 12)(10 11)(17 22)(18 21)(19 24)(20 23)
(5 10)(6 9)(11 15)(12 16)
(1 3)(2 4)(7 8)(13 14)
(1 3)(2 4)(5 10)(6 9)(7 8)(11 15)(12 16)(13 14)(17 19)(18 20)(21 23)(22 24)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 13)(2 8)(3 14)(4 7)(5 9 10 6)(11 12 15 16)(17 19)(21 23)
(1 18 6)(2 22 15)(3 20 9)(4 24 11)(5 14 17)(7 21 16)(8 23 12)(10 13 19)
G:=sub<Sym(24)| (1,7)(2,14)(3,8)(4,13)(5,15)(6,16)(9,12)(10,11)(17,22)(18,21)(19,24)(20,23), (5,10)(6,9)(11,15)(12,16), (1,3)(2,4)(7,8)(13,14), (1,3)(2,4)(5,10)(6,9)(7,8)(11,15)(12,16)(13,14)(17,19)(18,20)(21,23)(22,24), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,13)(2,8)(3,14)(4,7)(5,9,10,6)(11,12,15,16)(17,19)(21,23), (1,18,6)(2,22,15)(3,20,9)(4,24,11)(5,14,17)(7,21,16)(8,23,12)(10,13,19)>;
G:=Group( (1,7)(2,14)(3,8)(4,13)(5,15)(6,16)(9,12)(10,11)(17,22)(18,21)(19,24)(20,23), (5,10)(6,9)(11,15)(12,16), (1,3)(2,4)(7,8)(13,14), (1,3)(2,4)(5,10)(6,9)(7,8)(11,15)(12,16)(13,14)(17,19)(18,20)(21,23)(22,24), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,13)(2,8)(3,14)(4,7)(5,9,10,6)(11,12,15,16)(17,19)(21,23), (1,18,6)(2,22,15)(3,20,9)(4,24,11)(5,14,17)(7,21,16)(8,23,12)(10,13,19) );
G=PermutationGroup([[(1,7),(2,14),(3,8),(4,13),(5,15),(6,16),(9,12),(10,11),(17,22),(18,21),(19,24),(20,23)], [(5,10),(6,9),(11,15),(12,16)], [(1,3),(2,4),(7,8),(13,14)], [(1,3),(2,4),(5,10),(6,9),(7,8),(11,15),(12,16),(13,14),(17,19),(18,20),(21,23),(22,24)], [(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,13),(2,8),(3,14),(4,7),(5,9,10,6),(11,12,15,16),(17,19),(21,23)], [(1,18,6),(2,22,15),(3,20,9),(4,24,11),(5,14,17),(7,21,16),(8,23,12),(10,13,19)]])
G:=TransitiveGroup(24,460);
►On 24 points - transitive group 24T461
Generators in S24
(1 2)(3 16)(4 14)(5 6)(7 8)(9 10)(11 15)(12 13)(17 22)(18 21)(19 24)(20 23)
(1 6)(2 5)(7 9)(8 10)
(3 12)(4 11)(13 16)(14 15)
(1 6)(2 5)(3 12)(4 11)(7 9)(8 10)(13 16)(14 15)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 8 6 10)(2 9 5 7)(3 13)(4 15)(11 14)(12 16)(17 23)(18 22)(19 21)(20 24)
(1 13 17)(2 12 22)(3 24 5)(4 20 8)(6 16 19)(7 14 23)(9 15 21)(10 11 18)
G:=sub<Sym(24)| (1,2)(3,16)(4,14)(5,6)(7,8)(9,10)(11,15)(12,13)(17,22)(18,21)(19,24)(20,23), (1,6)(2,5)(7,9)(8,10), (3,12)(4,11)(13,16)(14,15), (1,6)(2,5)(3,12)(4,11)(7,9)(8,10)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,8,6,10)(2,9,5,7)(3,13)(4,15)(11,14)(12,16)(17,23)(18,22)(19,21)(20,24), (1,13,17)(2,12,22)(3,24,5)(4,20,8)(6,16,19)(7,14,23)(9,15,21)(10,11,18)>;
G:=Group( (1,2)(3,16)(4,14)(5,6)(7,8)(9,10)(11,15)(12,13)(17,22)(18,21)(19,24)(20,23), (1,6)(2,5)(7,9)(8,10), (3,12)(4,11)(13,16)(14,15), (1,6)(2,5)(3,12)(4,11)(7,9)(8,10)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,8,6,10)(2,9,5,7)(3,13)(4,15)(11,14)(12,16)(17,23)(18,22)(19,21)(20,24), (1,13,17)(2,12,22)(3,24,5)(4,20,8)(6,16,19)(7,14,23)(9,15,21)(10,11,18) );
G=PermutationGroup([[(1,2),(3,16),(4,14),(5,6),(7,8),(9,10),(11,15),(12,13),(17,22),(18,21),(19,24),(20,23)], [(1,6),(2,5),(7,9),(8,10)], [(3,12),(4,11),(13,16),(14,15)], [(1,6),(2,5),(3,12),(4,11),(7,9),(8,10),(13,16),(14,15),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,8,6,10),(2,9,5,7),(3,13),(4,15),(11,14),(12,16),(17,23),(18,22),(19,21),(20,24)], [(1,13,17),(2,12,22),(3,24,5),(4,20,8),(6,16,19),(7,14,23),(9,15,21),(10,11,18)]])
G:=TransitiveGroup(24,461);
►On 24 points - transitive group 24T462
Generators in S24
(1 4)(2 3)(5 14)(6 13)(7 11)(8 9)(10 16)(12 15)(17 18)(19 20)(21 22)(23 24)
(1 14)(2 13)(3 6)(4 5)
(7 16)(8 15)(9 12)(10 11)
(1 14)(2 13)(3 6)(4 5)(7 16)(8 15)(9 12)(10 11)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 4 14 5)(2 6 13 3)(7 12)(8 10)(9 16)(11 15)(17 21)(18 24)(19 23)(20 22)
(1 16 24)(2 12 17)(3 15 18)(4 10 23)(5 11 21)(6 8 20)(7 22 14)(9 19 13)
G:=sub<Sym(24)| (1,4)(2,3)(5,14)(6,13)(7,11)(8,9)(10,16)(12,15)(17,18)(19,20)(21,22)(23,24), (1,14)(2,13)(3,6)(4,5), (7,16)(8,15)(9,12)(10,11), (1,14)(2,13)(3,6)(4,5)(7,16)(8,15)(9,12)(10,11)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,4,14,5)(2,6,13,3)(7,12)(8,10)(9,16)(11,15)(17,21)(18,24)(19,23)(20,22), (1,16,24)(2,12,17)(3,15,18)(4,10,23)(5,11,21)(6,8,20)(7,22,14)(9,19,13)>;
G:=Group( (1,4)(2,3)(5,14)(6,13)(7,11)(8,9)(10,16)(12,15)(17,18)(19,20)(21,22)(23,24), (1,14)(2,13)(3,6)(4,5), (7,16)(8,15)(9,12)(10,11), (1,14)(2,13)(3,6)(4,5)(7,16)(8,15)(9,12)(10,11)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,4,14,5)(2,6,13,3)(7,12)(8,10)(9,16)(11,15)(17,21)(18,24)(19,23)(20,22), (1,16,24)(2,12,17)(3,15,18)(4,10,23)(5,11,21)(6,8,20)(7,22,14)(9,19,13) );
G=PermutationGroup([[(1,4),(2,3),(5,14),(6,13),(7,11),(8,9),(10,16),(12,15),(17,18),(19,20),(21,22),(23,24)], [(1,14),(2,13),(3,6),(4,5)], [(7,16),(8,15),(9,12),(10,11)], [(1,14),(2,13),(3,6),(4,5),(7,16),(8,15),(9,12),(10,11),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,4,14,5),(2,6,13,3),(7,12),(8,10),(9,16),(11,15),(17,21),(18,24),(19,23),(20,22)], [(1,16,24),(2,12,17),(3,15,18),(4,10,23),(5,11,21),(6,8,20),(7,22,14),(9,19,13)]])
G:=TransitiveGroup(24,462);
►On 24 points - transitive group 24T467
Generators in S24
(1 14)(3 8)(4 7)(10 15)(18 20)(22 24)
(3 8)(4 7)(5 12)(6 11)
(1 14)(2 13)(9 16)(10 15)
(1 14)(2 13)(3 8)(4 7)(5 12)(6 11)(9 16)(10 15)(17 19)(18 20)(21 23)(22 24)
(1 2)(3 4)(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 15)(2 9)(3 11 8 6)(4 5 7 12)(10 14)(13 16)(17 24)(18 23)(19 22)(20 21)
(1 18 7)(2 23 11)(3 15 24)(4 14 20)(5 9 19)(6 13 21)(8 10 22)(12 16 17)
G:=sub<Sym(24)| (1,14)(3,8)(4,7)(10,15)(18,20)(22,24), (3,8)(4,7)(5,12)(6,11), (1,14)(2,13)(9,16)(10,15), (1,14)(2,13)(3,8)(4,7)(5,12)(6,11)(9,16)(10,15)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,15)(2,9)(3,11,8,6)(4,5,7,12)(10,14)(13,16)(17,24)(18,23)(19,22)(20,21), (1,18,7)(2,23,11)(3,15,24)(4,14,20)(5,9,19)(6,13,21)(8,10,22)(12,16,17)>;
G:=Group( (1,14)(3,8)(4,7)(10,15)(18,20)(22,24), (3,8)(4,7)(5,12)(6,11), (1,14)(2,13)(9,16)(10,15), (1,14)(2,13)(3,8)(4,7)(5,12)(6,11)(9,16)(10,15)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,15)(2,9)(3,11,8,6)(4,5,7,12)(10,14)(13,16)(17,24)(18,23)(19,22)(20,21), (1,18,7)(2,23,11)(3,15,24)(4,14,20)(5,9,19)(6,13,21)(8,10,22)(12,16,17) );
G=PermutationGroup([[(1,14),(3,8),(4,7),(10,15),(18,20),(22,24)], [(3,8),(4,7),(5,12),(6,11)], [(1,14),(2,13),(9,16),(10,15)], [(1,14),(2,13),(3,8),(4,7),(5,12),(6,11),(9,16),(10,15),(17,19),(18,20),(21,23),(22,24)], [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,15),(2,9),(3,11,8,6),(4,5,7,12),(10,14),(13,16),(17,24),(18,23),(19,22),(20,21)], [(1,18,7),(2,23,11),(3,15,24),(4,14,20),(5,9,19),(6,13,21),(8,10,22),(12,16,17)]])
G:=TransitiveGroup(24,467);
►On 24 points - transitive group 24T468
Generators in S24
(1 3)(2 4)(5 14)(6 8)(7 10)(9 13)(11 15)(12 16)(17 23)(18 22)(19 21)(20 24)
(1 2)(3 4)(11 12)(15 16)
(5 10)(6 9)(7 14)(8 13)
(1 2)(3 4)(5 10)(6 9)(7 14)(8 13)(11 12)(15 16)(17 19)(18 20)(21 23)(22 24)
(5 6)(7 8)(9 10)(11 12)(13 14)(15 16)(17 18 19 20)(21 22 23 24)
(1 12 2 11)(3 15 4 16)(5 10)(7 14)(17 20)(18 19)(21 24)(22 23)
(1 14 20)(2 7 18)(3 5 24)(4 10 22)(6 21 16)(8 19 12)(9 23 15)(11 13 17)
G:=sub<Sym(24)| (1,3)(2,4)(5,14)(6,8)(7,10)(9,13)(11,15)(12,16)(17,23)(18,22)(19,21)(20,24), (1,2)(3,4)(11,12)(15,16), (5,10)(6,9)(7,14)(8,13), (1,2)(3,4)(5,10)(6,9)(7,14)(8,13)(11,12)(15,16)(17,19)(18,20)(21,23)(22,24), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,12,2,11)(3,15,4,16)(5,10)(7,14)(17,20)(18,19)(21,24)(22,23), (1,14,20)(2,7,18)(3,5,24)(4,10,22)(6,21,16)(8,19,12)(9,23,15)(11,13,17)>;
G:=Group( (1,3)(2,4)(5,14)(6,8)(7,10)(9,13)(11,15)(12,16)(17,23)(18,22)(19,21)(20,24), (1,2)(3,4)(11,12)(15,16), (5,10)(6,9)(7,14)(8,13), (1,2)(3,4)(5,10)(6,9)(7,14)(8,13)(11,12)(15,16)(17,19)(18,20)(21,23)(22,24), (5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18,19,20)(21,22,23,24), (1,12,2,11)(3,15,4,16)(5,10)(7,14)(17,20)(18,19)(21,24)(22,23), (1,14,20)(2,7,18)(3,5,24)(4,10,22)(6,21,16)(8,19,12)(9,23,15)(11,13,17) );
G=PermutationGroup([[(1,3),(2,4),(5,14),(6,8),(7,10),(9,13),(11,15),(12,16),(17,23),(18,22),(19,21),(20,24)], [(1,2),(3,4),(11,12),(15,16)], [(5,10),(6,9),(7,14),(8,13)], [(1,2),(3,4),(5,10),(6,9),(7,14),(8,13),(11,12),(15,16),(17,19),(18,20),(21,23),(22,24)], [(5,6),(7,8),(9,10),(11,12),(13,14),(15,16),(17,18,19,20),(21,22,23,24)], [(1,12,2,11),(3,15,4,16),(5,10),(7,14),(17,20),(18,19),(21,24),(22,23)], [(1,14,20),(2,7,18),(3,5,24),(4,10,22),(6,21,16),(8,19,12),(9,23,15),(11,13,17)]])
G:=TransitiveGroup(24,468);
Polynomial with Galois group C24.2A4 over ℚ
| action | f(x) | Disc(f) |
|---|---|---|
| 12T91 | x12-534x10+78489x8-4839186x6+143348046x4-2021896020x2+10884540241 | 236·316·1730·1930·4235574 |
| 12T93 | x12-30x10+327x8-1566x6+3096x4-1938x2+323 | 224·320·712·175·195 |
Matrix representation of C24.2A4 ►in GL6(ℤ)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| -1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | 0 | 0 | -1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | -1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | -1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | -1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | -1 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
G:=sub<GL(6,Integers())| [0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1,0,0,0,0,0,0,-1],[1,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,-1,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,-1],[0,0,0,0,1,0,0,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0] >;
C24.2A4 in GAP, Magma, Sage, TeX
C_2^4._2A_4
% in TeX
G:=Group("C2^4.2A4"); // GroupNames label
G:=SmallGroup(192,197);
// by ID
G=gap.SmallGroup(192,197);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,632,135,268,4371,934,521,304,2531,1524]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=g^3=1,e^2=g*b*g^-1=b*c*d,f^2=g*c*g^-1=b,a*b=b*a,a*c=c*a,a*d=d*a,e*a*e^-1=a*b*d,f*a*f^-1=a*c*d,a*g=g*a,b*c=c*b,b*d=d*b,b*e=e*b,b*f=f*b,c*d=d*c,c*e=e*c,c*f=f*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=b*e*f,g*f*g^-1=c*d*e>;
// generators/relations
Export
Subgroup lattice of C24.2A4 in TeX
Character table of C24.2A4 in TeX