direct product, metabelian, nilpotent (class 4), monomial, 2-elementary
Aliases: C3×C2≀C4, C24⋊3C12, C23⋊C4⋊1C6, (C23×C6)⋊1C4, C22⋊C4⋊1C12, C4.D4⋊5C6, (C2×C12).17D4, (C22×C6).1D4, C23.1(C3×D4), C22≀C2.1C6, C23.1(C2×C12), C6.32(C23⋊C4), (C6×D4).174C22, (C2×C4).1(C3×D4), (C3×C22⋊C4)⋊3C4, (C3×C23⋊C4)⋊7C2, (C2×D4).1(C2×C6), C2.6(C3×C23⋊C4), (C22×C6).8(C2×C4), (C3×C22≀C2).3C2, (C3×C4.D4)⋊12C2, (C2×C6).73(C22⋊C4), C22.10(C3×C22⋊C4), SmallGroup(192,157)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2≀C4
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f4=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, be=eb, fbf-1=bcde, cd=dc, ce=ec, fcf-1=cde, fdf-1=de=ed, ef=fe >
Subgroups: 258 in 94 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, M4(2), C2×D4, C2×D4, C24, C24, C2×C12, C2×C12, C3×D4, C22×C6, C22×C6, C23⋊C4, C4.D4, C22≀C2, C3×C22⋊C4, C3×C22⋊C4, C3×M4(2), C6×D4, C6×D4, C23×C6, C2≀C4, C3×C23⋊C4, C3×C4.D4, C3×C22≀C2, C3×C2≀C4
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, C12, C2×C6, C22⋊C4, C2×C12, C3×D4, C23⋊C4, C3×C22⋊C4, C2≀C4, C3×C23⋊C4, C3×C2≀C4
►On 24 points - transitive group 24T285
(1 7 19)(2 8 20)(3 5 17)(4 6 18)(9 16 21)(10 13 22)(11 14 23)(12 15 24)
(2 12)(8 15)(20 24)
(2 12)(3 9)(5 16)(8 15)(17 21)(20 24)
(2 12)(4 10)(6 13)(8 15)(18 22)(20 24)
(1 11)(2 12)(3 9)(4 10)(5 16)(6 13)(7 14)(8 15)(17 21)(18 22)(19 23)(20 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)
G:=sub<Sym(24)| (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(8,15)(20,24), (2,12)(3,9)(5,16)(8,15)(17,21)(20,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)>;
G:=Group( (1,7,19)(2,8,20)(3,5,17)(4,6,18)(9,16,21)(10,13,22)(11,14,23)(12,15,24), (2,12)(8,15)(20,24), (2,12)(3,9)(5,16)(8,15)(17,21)(20,24), (2,12)(4,10)(6,13)(8,15)(18,22)(20,24), (1,11)(2,12)(3,9)(4,10)(5,16)(6,13)(7,14)(8,15)(17,21)(18,22)(19,23)(20,24), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24) );
G=PermutationGroup([[(1,7,19),(2,8,20),(3,5,17),(4,6,18),(9,16,21),(10,13,22),(11,14,23),(12,15,24)], [(2,12),(8,15),(20,24)], [(2,12),(3,9),(5,16),(8,15),(17,21),(20,24)], [(2,12),(4,10),(6,13),(8,15),(18,22),(20,24)], [(1,11),(2,12),(3,9),(4,10),(5,16),(6,13),(7,14),(8,15),(17,21),(18,22),(19,23),(20,24)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)]])
G:=TransitiveGroup(24,285);
►On 24 points - transitive group 24T351
(1 9 12)(2 10 11)(3 6 8)(4 5 7)(13 21 20)(14 22 17)(15 23 18)(16 24 19)
(2 16)(4 17)(5 14)(7 22)(10 24)(11 19)
(1 13)(2 16)(3 18)(4 17)(5 14)(6 15)(7 22)(8 23)(9 21)(10 24)(11 19)(12 20)
(2 5)(4 11)(7 10)(14 16)(17 19)(22 24)
(1 6)(2 5)(3 12)(4 11)(7 10)(8 9)(13 15)(14 16)(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)
G:=sub<Sym(24)| (1,9,12)(2,10,11)(3,6,8)(4,5,7)(13,21,20)(14,22,17)(15,23,18)(16,24,19), (2,16)(4,17)(5,14)(7,22)(10,24)(11,19), (1,13)(2,16)(3,18)(4,17)(5,14)(6,15)(7,22)(8,23)(9,21)(10,24)(11,19)(12,20), (2,5)(4,11)(7,10)(14,16)(17,19)(22,24), (1,6)(2,5)(3,12)(4,11)(7,10)(8,9)(13,15)(14,16)(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)>;
G:=Group( (1,9,12)(2,10,11)(3,6,8)(4,5,7)(13,21,20)(14,22,17)(15,23,18)(16,24,19), (2,16)(4,17)(5,14)(7,22)(10,24)(11,19), (1,13)(2,16)(3,18)(4,17)(5,14)(6,15)(7,22)(8,23)(9,21)(10,24)(11,19)(12,20), (2,5)(4,11)(7,10)(14,16)(17,19)(22,24), (1,6)(2,5)(3,12)(4,11)(7,10)(8,9)(13,15)(14,16)(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) );
G=PermutationGroup([[(1,9,12),(2,10,11),(3,6,8),(4,5,7),(13,21,20),(14,22,17),(15,23,18),(16,24,19)], [(2,16),(4,17),(5,14),(7,22),(10,24),(11,19)], [(1,13),(2,16),(3,18),(4,17),(5,14),(6,15),(7,22),(8,23),(9,21),(10,24),(11,19),(12,20)], [(2,5),(4,11),(7,10),(14,16),(17,19),(22,24)], [(1,6),(2,5),(3,12),(4,11),(7,10),(8,9),(13,15),(14,16),(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)]])
G:=TransitiveGroup(24,351);
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 8A | 8B | 12A | 12B | 12C | ··· | 12H | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 4 | 4 | 8 | ··· | 8 | 8 | 8 | 8 | 8 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | ||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C12 | C12 | D4 | D4 | C3×D4 | C3×D4 | C23⋊C4 | C2≀C4 | C3×C23⋊C4 | C3×C2≀C4 |
| kernel | C3×C2≀C4 | C3×C23⋊C4 | C3×C4.D4 | C3×C22≀C2 | C2≀C4 | C3×C22⋊C4 | C23×C6 | C23⋊C4 | C4.D4 | C22≀C2 | C22⋊C4 | C24 | C2×C12 | C22×C6 | C2×C4 | C23 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 1 | 1 | 2 | 2 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C2≀C4 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 1 | 0 | 4 | 0 |
| 0 | 1 | 5 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 0 | 4 | 0 |
| 1 | 0 | 1 | 5 |
| 0 | 0 | 6 | 0 |
| 4 | 3 | 5 | 0 |
| 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 |
| 2 | 0 | 4 | 1 |
| 0 | 0 | 2 | 1 |
| 2 | 5 | 6 | 4 |
| 4 | 4 | 5 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[1,0,0,0,0,1,0,0,4,5,6,0,0,0,0,1],[1,1,0,4,0,0,0,3,4,1,6,5,0,5,0,0],[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],[2,0,2,4,0,0,5,4,4,2,6,5,1,1,4,6] >;
C3×C2≀C4 in GAP, Magma, Sage, TeX
C_3\times C_2\wr C_4
% in TeX
G:=Group("C3xC2wrC4"); // GroupNames label
G:=SmallGroup(192,157);
// by ID
G=gap.SmallGroup(192,157);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,1683,1271,375,6053]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=f^4=1,a*b=b*a,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,f*b*f^-1=b*c*d*e,c*d=d*c,c*e=e*c,f*c*f^-1=c*d*e,f*d*f^-1=d*e=e*d,e*f=f*e>;
// generators/relations