direct product, non-abelian, soluble
Aliases: C2×C23.3A4, C24.8A4, C23.4SL2(𝔽3), C23.9(C2×A4), C2.C42⋊3C6, C22.2(C42⋊C3), C22.1(C2×SL2(𝔽3)), C2.2(C2×C42⋊C3), (C2×C2.C42)⋊C3, SmallGroup(192,189)
Series: Derived ►Chief ►Lower central ►Upper central
| C2.C42 — C2×C23.3A4 |
Generators and relations for C2×C23.3A4
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, ae=ea, af=fa, 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 >
Subgroups: 327 in 75 conjugacy classes, 14 normal (10 characteristic)
C1, C2, C2, C2, C3, C4, C22, C22, C6, C2×C4, C23, C23, C23, A4, C2×C6, C22×C4, C24, C2×A4, C2.C42, C2.C42, C23×C4, C22×A4, C2×C2.C42, C23.3A4, C2×C23.3A4
Quotients: C1, C2, C3, C6, A4, SL2(𝔽3), C2×A4, C42⋊C3, C2×SL2(𝔽3), C23.3A4, C2×C42⋊C3, C2×C23.3A4
Character table of C2×C23.3A4
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 16 | 16 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 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 | 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 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 3 |
| ρ4 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ3 | ζ32 | ζ65 | ζ65 | ζ6 | ζ6 | linear of order 6 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 3 |
| ρ6 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | -1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | ζ32 | ζ3 | ζ6 | ζ6 | ζ65 | ζ65 | linear of order 6 |
| ρ7 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | -1 | -1 | 0 | 0 | 0 | 0 | 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 | 2 | -2 | -1 | -1 | 0 | 0 | 0 | 0 | 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 | 2 | -2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ65 | ζ3 | ζ6 | ζ32 | complex lifted from SL2(𝔽3) |
| ρ10 | 2 | 2 | -2 | -2 | 2 | -2 | 2 | -2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ6 | ζ32 | ζ65 | ζ3 | complex lifted from SL2(𝔽3) |
| ρ11 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | ζ3 | ζ32 | ζ6 | ζ3 | ζ65 | complex lifted from SL2(𝔽3) |
| ρ12 | 2 | -2 | -2 | 2 | -2 | -2 | 2 | 2 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | ζ32 | ζ3 | ζ65 | ζ32 | ζ6 | complex lifted from SL2(𝔽3) |
| ρ13 | 3 | -3 | 3 | -3 | -3 | 3 | 3 | -3 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×A4 |
| ρ14 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from A4 |
| ρ15 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | -1 | 1+2i | 1-2i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ16 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ17 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | 1+2i | -1 | -1 | 1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ18 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ19 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | -1+2i | 1 | 1 | -1-2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ20 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | -1-2i | 1 | 1 | -1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C42⋊C3 |
| ρ21 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | 1 | -1+2i | -1-2i | 1 | -1 | 1-2i | 1+2i | -1 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ22 | 3 | -3 | 3 | -3 | 1 | -1 | -1 | 1 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | 1-2i | -1 | -1 | 1+2i | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C2×C42⋊C3 |
| ρ23 | 6 | 6 | -6 | -6 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.3A4 |
| ρ24 | 6 | -6 | -6 | 6 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C23.3A4 |
►On 24 points - transitive group 24T422
Generators in S24
(1 2)(3 8)(4 7)(5 6)(9 11)(10 12)(13 23)(14 24)(15 21)(16 22)(17 19)(18 20)
(1 2)(3 8)(4 7)(5 6)(9 19)(10 20)(11 17)(12 18)
(1 5)(2 6)(3 4)(7 8)(13 23)(14 24)(15 21)(16 22)
(1 6)(2 5)(3 7)(4 8)(9 17)(10 18)(11 19)(12 20)(13 21)(14 22)(15 23)(16 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 2 7)(3 6 8 5)(9 20 19 10)(11 18 17 12)(13 21)(15 23)
(1 16 18)(2 22 20)(3 15 17)(4 13 11)(5 14 12)(6 24 10)(7 23 9)(8 21 19)
G:=sub<Sym(24)| (1,2)(3,8)(4,7)(5,6)(9,11)(10,12)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20), (1,2)(3,8)(4,7)(5,6)(9,19)(10,20)(11,17)(12,18), (1,5)(2,6)(3,4)(7,8)(13,23)(14,24)(15,21)(16,22), (1,6)(2,5)(3,7)(4,8)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,7)(3,6,8,5)(9,20,19,10)(11,18,17,12)(13,21)(15,23), (1,16,18)(2,22,20)(3,15,17)(4,13,11)(5,14,12)(6,24,10)(7,23,9)(8,21,19)>;
G:=Group( (1,2)(3,8)(4,7)(5,6)(9,11)(10,12)(13,23)(14,24)(15,21)(16,22)(17,19)(18,20), (1,2)(3,8)(4,7)(5,6)(9,19)(10,20)(11,17)(12,18), (1,5)(2,6)(3,4)(7,8)(13,23)(14,24)(15,21)(16,22), (1,6)(2,5)(3,7)(4,8)(9,17)(10,18)(11,19)(12,20)(13,21)(14,22)(15,23)(16,24), (1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,4,2,7)(3,6,8,5)(9,20,19,10)(11,18,17,12)(13,21)(15,23), (1,16,18)(2,22,20)(3,15,17)(4,13,11)(5,14,12)(6,24,10)(7,23,9)(8,21,19) );
G=PermutationGroup([[(1,2),(3,8),(4,7),(5,6),(9,11),(10,12),(13,23),(14,24),(15,21),(16,22),(17,19),(18,20)], [(1,2),(3,8),(4,7),(5,6),(9,19),(10,20),(11,17),(12,18)], [(1,5),(2,6),(3,4),(7,8),(13,23),(14,24),(15,21),(16,22)], [(1,6),(2,5),(3,7),(4,8),(9,17),(10,18),(11,19),(12,20),(13,21),(14,22),(15,23),(16,24)], [(1,2),(3,4),(5,6),(7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,4,2,7),(3,6,8,5),(9,20,19,10),(11,18,17,12),(13,21),(15,23)], [(1,16,18),(2,22,20),(3,15,17),(4,13,11),(5,14,12),(6,24,10),(7,23,9),(8,21,19)]])
G:=TransitiveGroup(24,422);
►On 24 points - transitive group 24T423
Generators in S24
(1 2)(3 5)(4 6)(7 8)(9 12)(10 11)(13 16)(14 15)(17 22)(18 23)(19 24)(20 21)
(9 16)(10 15)(11 14)(12 13)
(1 7)(2 8)(3 6)(4 5)
(1 7)(2 8)(3 6)(4 5)(9 16)(10 15)(11 14)(12 13)(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 3)(2 5)(4 8)(6 7)(9 11 16 14)(10 13 15 12)(17 22)(18 21)(19 24)(20 23)
(1 17 13)(2 22 16)(3 23 11)(4 20 15)(5 18 10)(6 21 14)(7 19 12)(8 24 9)
G:=sub<Sym(24)| (1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,22)(18,23)(19,24)(20,21), (9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,6)(4,5), (1,7)(2,8)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(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), (1,3)(2,5)(4,8)(6,7)(9,11,16,14)(10,13,15,12)(17,22)(18,21)(19,24)(20,23), (1,17,13)(2,22,16)(3,23,11)(4,20,15)(5,18,10)(6,21,14)(7,19,12)(8,24,9)>;
G:=Group( (1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,22)(18,23)(19,24)(20,21), (9,16)(10,15)(11,14)(12,13), (1,7)(2,8)(3,6)(4,5), (1,7)(2,8)(3,6)(4,5)(9,16)(10,15)(11,14)(12,13)(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), (1,3)(2,5)(4,8)(6,7)(9,11,16,14)(10,13,15,12)(17,22)(18,21)(19,24)(20,23), (1,17,13)(2,22,16)(3,23,11)(4,20,15)(5,18,10)(6,21,14)(7,19,12)(8,24,9) );
G=PermutationGroup([[(1,2),(3,5),(4,6),(7,8),(9,12),(10,11),(13,16),(14,15),(17,22),(18,23),(19,24),(20,21)], [(9,16),(10,15),(11,14),(12,13)], [(1,7),(2,8),(3,6),(4,5)], [(1,7),(2,8),(3,6),(4,5),(9,16),(10,15),(11,14),(12,13),(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)], [(1,3),(2,5),(4,8),(6,7),(9,11,16,14),(10,13,15,12),(17,22),(18,21),(19,24),(20,23)], [(1,17,13),(2,22,16),(3,23,11),(4,20,15),(5,18,10),(6,21,14),(7,19,12),(8,24,9)]])
G:=TransitiveGroup(24,423);
►On 24 points - transitive group 24T424
Generators in S24
(1 4)(2 3)(5 9)(6 10)(7 13)(8 14)(11 16)(12 15)(17 24)(18 21)(19 22)(20 23)
(1 2)(3 4)(7 8)(13 14)
(5 11)(6 12)(9 16)(10 15)
(1 2)(3 4)(5 11)(6 12)(7 8)(9 16)(10 15)(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 14 2 13)(3 7 4 8)(5 11)(9 16)(17 18)(19 20)(21 24)(22 23)
(1 5 22)(2 11 24)(3 16 17)(4 9 19)(6 23 13)(7 10 20)(8 15 18)(12 21 14)
G:=sub<Sym(24)| (1,4)(2,3)(5,9)(6,10)(7,13)(8,14)(11,16)(12,15)(17,24)(18,21)(19,22)(20,23), (1,2)(3,4)(7,8)(13,14), (5,11)(6,12)(9,16)(10,15), (1,2)(3,4)(5,11)(6,12)(7,8)(9,16)(10,15)(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), (1,14,2,13)(3,7,4,8)(5,11)(9,16)(17,18)(19,20)(21,24)(22,23), (1,5,22)(2,11,24)(3,16,17)(4,9,19)(6,23,13)(7,10,20)(8,15,18)(12,21,14)>;
G:=Group( (1,4)(2,3)(5,9)(6,10)(7,13)(8,14)(11,16)(12,15)(17,24)(18,21)(19,22)(20,23), (1,2)(3,4)(7,8)(13,14), (5,11)(6,12)(9,16)(10,15), (1,2)(3,4)(5,11)(6,12)(7,8)(9,16)(10,15)(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), (1,14,2,13)(3,7,4,8)(5,11)(9,16)(17,18)(19,20)(21,24)(22,23), (1,5,22)(2,11,24)(3,16,17)(4,9,19)(6,23,13)(7,10,20)(8,15,18)(12,21,14) );
G=PermutationGroup([[(1,4),(2,3),(5,9),(6,10),(7,13),(8,14),(11,16),(12,15),(17,24),(18,21),(19,22),(20,23)], [(1,2),(3,4),(7,8),(13,14)], [(5,11),(6,12),(9,16),(10,15)], [(1,2),(3,4),(5,11),(6,12),(7,8),(9,16),(10,15),(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)], [(1,14,2,13),(3,7,4,8),(5,11),(9,16),(17,18),(19,20),(21,24),(22,23)], [(1,5,22),(2,11,24),(3,16,17),(4,9,19),(6,23,13),(7,10,20),(8,15,18),(12,21,14)]])
G:=TransitiveGroup(24,424);
Matrix representation of C2×C23.3A4 ►in GL5(𝔽13)
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 8 |
| 9 | 10 | 0 | 0 | 0 |
| 10 | 4 | 0 | 0 | 0 |
| 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 12 |
| 3 | 12 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 | 0 |
G:=sub<GL(5,GF(13))| [12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1],[12,0,0,0,0,0,12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,12],[12,0,0,0,0,0,12,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1],[0,12,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,5,0,0,0,0,0,8],[9,10,0,0,0,10,4,0,0,0,0,0,5,0,0,0,0,0,5,0,0,0,0,0,12],[3,0,0,0,0,12,9,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,1,0] >;
C2×C23.3A4 in GAP, Magma, Sage, TeX
C_2\times C_2^3._3A_4
% in TeX
G:=Group("C2xC2^3.3A4"); // GroupNames label
G:=SmallGroup(192,189);
// by ID
G=gap.SmallGroup(192,189);
# by ID
G:=PCGroup([7,-2,-3,-2,2,-2,2,-2,135,268,934,521,80,2531,3540]);
// 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,a*e=e*a,a*f=f*a,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
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