direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C22≀C2, C24⋊5C6, (C2×C6)⋊7D4, (C2×D4)⋊1C6, C2.4(C6×D4), (C6×D4)⋊10C2, C22⋊C4⋊2C6, (C23×C6)⋊1C2, C23⋊1(C2×C6), C6.67(C2×D4), C22⋊3(C3×D4), (C2×C12)⋊8C22, (C2×C6).75C23, (C22×C6)⋊1C22, C22.10(C22×C6), (C2×C4)⋊1(C2×C6), (C3×C22⋊C4)⋊10C2, SmallGroup(96,167)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C22≀C2
G = < a,b,c,d,e,f | a3=b2=c2=d2=e2=f2=1, ab=ba, ac=ca, ad=da, ae=ea, af=fa, bc=cb, fbf=bd=db, be=eb, cd=dc, fcf=ce=ec, de=ed, df=fd, ef=fe >
Subgroups: 212 in 130 conjugacy classes, 52 normal (10 characteristic)
C1, C2, C2, C3, C4, C22, C22, C22, C6, C6, C2×C4, D4, C23, C23, C23, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C2×D4, C24, C2×C12, C3×D4, C22×C6, C22×C6, C22×C6, C22≀C2, C3×C22⋊C4, C6×D4, C23×C6, C3×C22≀C2
Quotients: C1, C2, C3, C22, C6, D4, C23, C2×C6, C2×D4, C3×D4, C22×C6, C22≀C2, C6×D4, C3×C22≀C2
►On 24 points - transitive group 24T112
(1 2 3)(4 5 6)(7 8 9)(10 11 12)(13 14 15)(16 17 18)(19 20 21)(22 23 24)
(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)
(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)
(1 15)(2 13)(3 14)(4 20)(5 21)(6 19)(7 24)(8 22)(9 23)(10 18)(11 16)(12 17)
(1 10)(2 11)(3 12)(4 22)(5 23)(6 24)(7 19)(8 20)(9 21)(13 16)(14 17)(15 18)
(1 4)(2 5)(3 6)(7 17)(8 18)(9 16)(10 22)(11 23)(12 24)(13 21)(14 19)(15 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), (4,20)(5,21)(6,19)(7,24)(8,22)(9,23), (4,22)(5,23)(6,24)(7,19)(8,20)(9,21), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,10)(2,11)(3,12)(4,22)(5,23)(6,24)(7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20)>;
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), (4,20)(5,21)(6,19)(7,24)(8,22)(9,23), (4,22)(5,23)(6,24)(7,19)(8,20)(9,21), (1,15)(2,13)(3,14)(4,20)(5,21)(6,19)(7,24)(8,22)(9,23)(10,18)(11,16)(12,17), (1,10)(2,11)(3,12)(4,22)(5,23)(6,24)(7,19)(8,20)(9,21)(13,16)(14,17)(15,18), (1,4)(2,5)(3,6)(7,17)(8,18)(9,16)(10,22)(11,23)(12,24)(13,21)(14,19)(15,20) );
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)], [(4,20),(5,21),(6,19),(7,24),(8,22),(9,23)], [(4,22),(5,23),(6,24),(7,19),(8,20),(9,21)], [(1,15),(2,13),(3,14),(4,20),(5,21),(6,19),(7,24),(8,22),(9,23),(10,18),(11,16),(12,17)], [(1,10),(2,11),(3,12),(4,22),(5,23),(6,24),(7,19),(8,20),(9,21),(13,16),(14,17),(15,18)], [(1,4),(2,5),(3,6),(7,17),(8,18),(9,16),(10,22),(11,23),(12,24),(13,21),(14,19),(15,20)]])
G:=TransitiveGroup(24,112);
C3×C22≀C2 is a maximal subgroup of
C24⋊5Dic3 C24⋊6D6 C24.67D6 C24.43D6 C24⋊7D6 C24⋊8D6 C24.44D6 C24.45D6 C24.46D6 C24⋊9D6 C24.47D6 C3×D42 C24⋊C18
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | ··· | 2I | 2J | 3A | 3B | 4A | 4B | 4C | 6A | ··· | 6F | 6G | ··· | 6R | 6S | 6T | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | 1 | 1 | 4 | 4 | 4 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 |
| type | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | D4 | C3×D4 |
| kernel | C3×C22≀C2 | C3×C22⋊C4 | C6×D4 | C23×C6 | C22≀C2 | C22⋊C4 | C2×D4 | C24 | C2×C6 | C22 |
| # reps | 1 | 3 | 3 | 1 | 2 | 6 | 6 | 2 | 6 | 12 |
Matrix representation of C3×C22≀C2 ►in GL4(𝔽13) generated by
| 9 | 0 | 0 | 0 |
| 0 | 9 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(13))| [9,0,0,0,0,9,0,0,0,0,9,0,0,0,0,9],[12,0,0,0,0,1,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,12,0,0,0,0,12,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,1,0,0,0,0,1],[12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12],[0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0] >;
C3×C22≀C2 in GAP, Magma, Sage, TeX
C_3\times C_2^2\wr C_2
% in TeX
G:=Group("C3xC2^2wrC2"); // GroupNames label
G:=SmallGroup(96,167);
// by ID
G=gap.SmallGroup(96,167);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-2,313,938]);
// Polycyclic
G:=Group<a,b,c,d,e,f|a^3=b^2=c^2=d^2=e^2=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,f*b*f=b*d=d*b,b*e=e*b,c*d=d*c,f*c*f=c*e=e*c,d*e=e*d,d*f=f*d,e*f=f*e>;
// generators/relations