direct product, non-abelian, soluble, monomial
Aliases: C3×C23⋊A4, 2+ 1+4⋊3C32, Q8⋊3(C3×A4), (C3×Q8)⋊2A4, C23⋊2(C3×A4), (C22×C6)⋊2A4, C6.5(C22⋊A4), (C3×2+ 1+4)⋊2C3, C2.2(C3×C22⋊A4), SmallGroup(288,987)
Series: Derived ►Chief ►Lower central ►Upper central
| 2+ 1+4 — C3×C23⋊A4 |
Generators and relations for C3×C23⋊A4
G = < a,b,c,d,e,f,g | a3=b2=c2=d2=e2=f2=g3=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, ag=ga, gbg-1=bc=cb, fbf=bd=db, be=eb, ece=cd=dc, cf=fc, gcg-1=b, de=ed, df=fd, dg=gd, geg-1=ef=fe, gfg-1=e >
Subgroups: 480 in 114 conjugacy classes, 20 normal (7 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C2×C4, D4, Q8, C23, C23, C32, C12, A4, C2×C6, C2×D4, C4○D4, C3×C6, SL2(𝔽3), C2×C12, C3×D4, C3×Q8, C2×A4, C22×C6, C22×C6, 2+ 1+4, C3×A4, C6×D4, C3×C4○D4, C3×SL2(𝔽3), C6×A4, C23⋊A4, C3×2+ 1+4, C3×C23⋊A4
Quotients: C1, C3, C32, A4, C3×A4, C22⋊A4, C23⋊A4, C3×C22⋊A4, C3×C23⋊A4
►On 24 points - transitive group 24T591
(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)(2 9)(3 7)(4 23)(5 24)(6 22)(10 20)(11 21)(12 19)(13 18)(14 16)(15 17)
(1 12)(2 10)(3 11)(4 16)(5 17)(6 18)(7 21)(8 19)(9 20)(13 22)(14 23)(15 24)
(1 23)(2 24)(3 22)(4 8)(5 9)(6 7)(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(10 15)(11 13)(12 14)(16 19)(17 20)(18 21)
(4 8)(5 9)(6 7)(16 19)(17 20)(18 21)
(4 14 16)(5 15 17)(6 13 18)(7 11 21)(8 12 19)(9 10 20)
G:=sub<Sym(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)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (4,8)(5,9)(6,7)(16,19)(17,20)(18,21), (4,14,16)(5,15,17)(6,13,18)(7,11,21)(8,12,19)(9,10,20)>;
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), (1,8)(2,9)(3,7)(4,23)(5,24)(6,22)(10,20)(11,21)(12,19)(13,18)(14,16)(15,17), (1,12)(2,10)(3,11)(4,16)(5,17)(6,18)(7,21)(8,19)(9,20)(13,22)(14,23)(15,24), (1,23)(2,24)(3,22)(4,8)(5,9)(6,7)(10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (10,15)(11,13)(12,14)(16,19)(17,20)(18,21), (4,8)(5,9)(6,7)(16,19)(17,20)(18,21), (4,14,16)(5,15,17)(6,13,18)(7,11,21)(8,12,19)(9,10,20) );
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)], [(1,8),(2,9),(3,7),(4,23),(5,24),(6,22),(10,20),(11,21),(12,19),(13,18),(14,16),(15,17)], [(1,12),(2,10),(3,11),(4,16),(5,17),(6,18),(7,21),(8,19),(9,20),(13,22),(14,23),(15,24)], [(1,23),(2,24),(3,22),(4,8),(5,9),(6,7),(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(10,15),(11,13),(12,14),(16,19),(17,20),(18,21)], [(4,8),(5,9),(6,7),(16,19),(17,20),(18,21)], [(4,14,16),(5,15,17),(6,13,18),(7,11,21),(8,12,19),(9,10,20)]])
G:=TransitiveGroup(24,591);
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | ··· | 3H | 4A | 4B | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6N | 12A | 12B | 12C | 12D |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | ··· | 3 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | 6 | 6 | 1 | 1 | 16 | ··· | 16 | 6 | 6 | 1 | 1 | 6 | ··· | 6 | 16 | ··· | 16 | 6 | 6 | 6 | 6 |
33 irreducible representations
| dim | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 4 | 4 | 4 |
| type | + | + | + | + | ||||||
| image | C1 | C3 | C3 | A4 | A4 | C3×A4 | C3×A4 | C23⋊A4 | C23⋊A4 | C3×C23⋊A4 |
| kernel | C3×C23⋊A4 | C23⋊A4 | C3×2+ 1+4 | C3×Q8 | C22×C6 | Q8 | C23 | C3 | C3 | C1 |
| # reps | 1 | 6 | 2 | 2 | 3 | 4 | 6 | 1 | 2 | 6 |
Matrix representation of C3×C23⋊A4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 4 | 5 | 2 | 6 |
| 0 | 6 | 0 | 2 |
| 3 | 3 | 3 | 1 |
| 0 | 0 | 0 | 1 |
| 0 | 6 | 3 | 2 |
| 6 | 0 | 4 | 2 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 5 | 2 | 3 | 2 |
| 3 | 2 | 4 | 2 |
| 2 | 5 | 6 | 4 |
| 3 | 3 | 4 | 1 |
| 0 | 1 | 1 | 5 |
| 0 | 1 | 5 | 0 |
| 0 | 0 | 6 | 0 |
| 3 | 4 | 2 | 0 |
| 2 | 3 | 2 | 4 |
| 0 | 5 | 4 | 4 |
| 0 | 6 | 0 | 1 |
| 0 | 0 | 0 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[4,0,3,0,5,6,3,0,2,0,3,0,6,2,1,1],[0,6,0,0,6,0,0,0,3,4,6,0,2,2,0,1],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[5,3,2,3,2,2,5,3,3,4,6,4,2,2,4,1],[0,0,0,3,1,1,0,4,1,5,6,2,5,0,0,0],[2,0,0,0,3,5,6,0,2,4,0,0,4,4,1,2] >;
C3×C23⋊A4 in GAP, Magma, Sage, TeX
C_3\times C_2^3\rtimes A_4
% in TeX
G:=Group("C3xC2^3:A4"); // GroupNames label
G:=SmallGroup(288,987);
// by ID
G=gap.SmallGroup(288,987);
# by ID
G:=PCGroup([7,-3,-3,-2,2,-2,2,-2,380,759,2524,375,4541,1027]);
// Polycyclic
G:=Group<a,b,c,d,e,f,g|a^3=b^2=c^2=d^2=e^2=f^2=g^3=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,a*f=f*a,a*g=g*a,g*b*g^-1=b*c=c*b,f*b*f=b*d=d*b,b*e=e*b,e*c*e=c*d=d*c,c*f=f*c,g*c*g^-1=b,d*e=e*d,d*f=f*d,d*g=g*d,g*e*g^-1=e*f=f*e,g*f*g^-1=e>;
// generators/relations