direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C2≀C22, 2+ 1+4⋊7C6, C23⋊(C3×D4), (C2×C12)⋊4D4, C23⋊C4⋊3C6, C24⋊4(C2×C6), (C22×C6)⋊1D4, C22≀C2⋊3C6, (C23×C6)⋊1C22, C22.18(C6×D4), C6.104C22≀C2, C23.3(C22×C6), (C6×D4).183C22, (C22×C6).82C23, (C3×2+ 1+4)⋊8C2, (C2×C4)⋊(C3×D4), (C3×C23⋊C4)⋊9C2, C22⋊C4⋊1(C2×C6), (C2×D4).8(C2×C6), (C2×C6).413(C2×D4), (C3×C22≀C2)⋊11C2, C2.18(C3×C22≀C2), (C3×C22⋊C4)⋊36C22, SmallGroup(192,890)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2≀C22
G = < a,b,c,d,e,f | a3=b2=c2=d2=e4=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, bd=db, ebe-1=fbf=bcd, ece-1=cd=dc, cf=fc, de=ed, df=fd, fef=e-1 >
Subgroups: 450 in 198 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C3, C4, C22, C22, C6, C6, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C12, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C2×D4, C4○D4, C24, C2×C12, C2×C12, C3×D4, C3×Q8, C22×C6, C22×C6, C22×C6, C23⋊C4, C22≀C2, 2+ 1+4, C3×C22⋊C4, C3×C22⋊C4, C6×D4, C6×D4, C3×C4○D4, C23×C6, C2≀C22, C3×C23⋊C4, C3×C22≀C2, C3×2+ 1+4, C3×C2≀C22
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, C2≀C22, C3×C22≀C2, C3×C2≀C22
►On 24 points - transitive group 24T286
(1 7 3)(2 8 4)(5 10 11)(6 9 12)(13 17 22)(14 18 23)(15 19 24)(16 20 21)
(1 19)(2 18)(3 15)(4 14)(5 17)(6 20)(7 24)(8 23)(9 21)(10 22)(11 13)(12 16)
(2 6)(4 12)(8 9)(14 16)(18 20)(21 23)
(1 5)(2 6)(3 11)(4 12)(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)
(13 15)(17 19)(22 24)
G:=sub<Sym(24)| (1,7,3)(2,8,4)(5,10,11)(6,9,12)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (1,19)(2,18)(3,15)(4,14)(5,17)(6,20)(7,24)(8,23)(9,21)(10,22)(11,13)(12,16), (2,6)(4,12)(8,9)(14,16)(18,20)(21,23), (1,5)(2,6)(3,11)(4,12)(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), (13,15)(17,19)(22,24)>;
G:=Group( (1,7,3)(2,8,4)(5,10,11)(6,9,12)(13,17,22)(14,18,23)(15,19,24)(16,20,21), (1,19)(2,18)(3,15)(4,14)(5,17)(6,20)(7,24)(8,23)(9,21)(10,22)(11,13)(12,16), (2,6)(4,12)(8,9)(14,16)(18,20)(21,23), (1,5)(2,6)(3,11)(4,12)(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), (13,15)(17,19)(22,24) );
G=PermutationGroup([[(1,7,3),(2,8,4),(5,10,11),(6,9,12),(13,17,22),(14,18,23),(15,19,24),(16,20,21)], [(1,19),(2,18),(3,15),(4,14),(5,17),(6,20),(7,24),(8,23),(9,21),(10,22),(11,13),(12,16)], [(2,6),(4,12),(8,9),(14,16),(18,20),(21,23)], [(1,5),(2,6),(3,11),(4,12),(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)], [(13,15),(17,19),(22,24)]])
G:=TransitiveGroup(24,286);
►On 24 points - transitive group 24T352
(1 15 21)(2 16 22)(3 13 23)(4 14 24)(5 9 20)(6 10 17)(7 11 18)(8 12 19)
(1 5)(2 8)(3 4)(6 7)(9 15)(10 11)(12 16)(13 14)(17 18)(19 22)(20 21)(23 24)
(1 3)(2 7)(4 5)(6 8)(9 14)(10 12)(11 16)(13 15)(17 19)(18 22)(20 24)(21 23)
(1 8)(2 5)(3 6)(4 7)(9 16)(10 13)(11 14)(12 15)(17 23)(18 24)(19 21)(20 22)
(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 7)(3 6)(4 5)(9 14)(10 13)(11 16)(12 15)(17 23)(18 22)(19 21)(20 24)
G:=sub<Sym(24)| (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19), (1,5)(2,8)(3,4)(6,7)(9,15)(10,11)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24), (1,3)(2,7)(4,5)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (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)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24)>;
G:=Group( (1,15,21)(2,16,22)(3,13,23)(4,14,24)(5,9,20)(6,10,17)(7,11,18)(8,12,19), (1,5)(2,8)(3,4)(6,7)(9,15)(10,11)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24), (1,3)(2,7)(4,5)(6,8)(9,14)(10,12)(11,16)(13,15)(17,19)(18,22)(20,24)(21,23), (1,8)(2,5)(3,6)(4,7)(9,16)(10,13)(11,14)(12,15)(17,23)(18,24)(19,21)(20,22), (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)(2,7)(3,6)(4,5)(9,14)(10,13)(11,16)(12,15)(17,23)(18,22)(19,21)(20,24) );
G=PermutationGroup([[(1,15,21),(2,16,22),(3,13,23),(4,14,24),(5,9,20),(6,10,17),(7,11,18),(8,12,19)], [(1,5),(2,8),(3,4),(6,7),(9,15),(10,11),(12,16),(13,14),(17,18),(19,22),(20,21),(23,24)], [(1,3),(2,7),(4,5),(6,8),(9,14),(10,12),(11,16),(13,15),(17,19),(18,22),(20,24),(21,23)], [(1,8),(2,5),(3,6),(4,7),(9,16),(10,13),(11,14),(12,15),(17,23),(18,24),(19,21),(20,22)], [(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),(2,7),(3,6),(4,5),(9,14),(10,13),(11,16),(12,15),(17,23),(18,22),(19,21),(20,24)]])
G:=TransitiveGroup(24,352);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2I | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | ··· | 6H | 6I | ··· | 6R | 12A | ··· | 12F | 12G | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 2 | 2 | 2 | 4 | ··· | 4 | 1 | 1 | 4 | 4 | 4 | 8 | 8 | 8 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 4 | ··· | 4 | 8 | ··· | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | D4 | C3×D4 | C3×D4 | C2≀C22 | C3×C2≀C22 |
| kernel | C3×C2≀C22 | C3×C23⋊C4 | C3×C22≀C2 | C3×2+ 1+4 | C2≀C22 | C23⋊C4 | C22≀C2 | 2+ 1+4 | C2×C12 | C22×C6 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 3 | 3 | 6 | 6 | 2 | 4 |
Matrix representation of C3×C2≀C22 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 4 | 3 | 2 | 0 |
| 5 | 0 | 3 | 1 |
| 6 | 1 | 2 | 2 |
| 4 | 4 | 5 | 1 |
| 4 | 3 | 1 | 0 |
| 4 | 1 | 4 | 3 |
| 1 | 6 | 5 | 5 |
| 6 | 2 | 0 | 4 |
| 6 | 0 | 0 | 0 |
| 0 | 6 | 0 | 0 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 6 | 4 | 2 |
| 3 | 6 | 0 | 1 |
| 6 | 5 | 2 | 6 |
| 6 | 1 | 4 | 5 |
| 6 | 5 | 2 | 1 |
| 0 | 4 | 2 | 1 |
| 0 | 5 | 1 | 1 |
| 0 | 3 | 4 | 1 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[4,5,6,4,3,0,1,4,2,3,2,5,0,1,2,1],[4,4,1,6,3,1,6,2,1,4,5,0,0,3,5,4],[6,0,0,0,0,6,0,0,0,0,6,0,0,0,0,6],[1,3,6,6,6,6,5,1,4,0,2,4,2,1,6,5],[6,0,0,0,5,4,5,3,2,2,1,4,1,1,1,1] >;
C3×C2≀C22 in GAP, Magma, Sage, TeX
C_3\times C_2\wr C_2^2
% in TeX
G:=Group("C3xC2wrC2^2"); // GroupNames label
G:=SmallGroup(192,890);
// by ID
G=gap.SmallGroup(192,890);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,1068,3036]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^4=f^2=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,e*b*e^-1=f*b*f=b*c*d,e*c*e^-1=c*d=d*c,c*f=f*c,d*e=e*d,d*f=f*d,f*e*f=e^-1>;
// generators/relations