direct product, metabelian, nilpotent (class 3), monomial, 2-elementary
Aliases: C3xD4:4D4, 2+ 1+4:6C6, C4wrC2:1C6, D4:4(C3xD4), Q8:5(C3xD4), (C3xD4):22D4, C4:1D4:3C6, C8:C22:1C6, C42:5(C2xC6), (C3xQ8):22D4, C4.27(C6xD4), C4.D4:1C6, C23.5(C3xD4), (C22xC6).5D4, (C4xC12):35C22, C12.388(C2xD4), (C6xD4):28C22, M4(2):1(C2xC6), C22.14(C6xD4), C6.100C22wrC2, (C2xC12).609C23, (C3x2+ 1+4):7C2, (C3xM4(2)):14C22, (C3xC4wrC2):5C2, (C2xD4):2(C2xC6), (C3xC8:C22):8C2, C4oD4.6(C2xC6), (C3xC4:1D4):11C2, (C2xC6).409(C2xD4), (C3xC4.D4):5C2, (C2xC4).4(C22xC6), C2.14(C3xC22wrC2), (C3xC4oD4).31C22, SmallGroup(192,886)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3xD4:4D4
G = < a,b,c,d,e | a3=b4=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b-1c, ece=bc, ede=d-1 >
Subgroups: 370 in 168 conjugacy classes, 54 normal (22 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, Q8, C23, C23, C12, C12, C2xC6, C2xC6, C42, M4(2), D8, SD16, C2xD4, C2xD4, C4oD4, C4oD4, C24, C2xC12, C2xC12, C3xD4, C3xD4, C3xQ8, C22xC6, C22xC6, C4.D4, C4wrC2, C4:1D4, C8:C22, 2+ 1+4, C4xC12, C3xM4(2), C3xD8, C3xSD16, C6xD4, C6xD4, C3xC4oD4, C3xC4oD4, D4:4D4, C3xC4.D4, C3xC4wrC2, C3xC4:1D4, C3xC8:C22, C3x2+ 1+4, C3xD4:4D4
Quotients: C1, C2, C3, C22, C6, D4, C23, C2xC6, C2xD4, C3xD4, C22xC6, C22wrC2, C6xD4, D4:4D4, C3xC22wrC2, C3xD4:4D4
►On 24 points - transitive group 24T350
(1 11 7)(2 12 8)(3 9 5)(4 10 6)(13 21 17)(14 22 18)(15 23 19)(16 24 20)
(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 16)(2 15)(3 14)(4 13)(5 18)(6 17)(7 20)(8 19)(9 22)(10 21)(11 24)(12 23)
(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)
(1 3)(5 7)(9 11)(13 16)(14 15)(17 20)(18 19)(21 24)(22 23)
G:=sub<Sym(24)| (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(5,7)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)>;
G:=Group( (1,11,7)(2,12,8)(3,9,5)(4,10,6)(13,21,17)(14,22,18)(15,23,19)(16,24,20), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,16)(2,15)(3,14)(4,13)(5,18)(6,17)(7,20)(8,19)(9,22)(10,21)(11,24)(12,23), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,14,15,16)(17,18,19,20)(21,22,23,24), (1,3)(5,7)(9,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23) );
G=PermutationGroup([[(1,11,7),(2,12,8),(3,9,5),(4,10,6),(13,21,17),(14,22,18),(15,23,19),(16,24,20)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,16),(2,15),(3,14),(4,13),(5,18),(6,17),(7,20),(8,19),(9,22),(10,21),(11,24),(12,23)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,14,15,16),(17,18,19,20),(21,22,23,24)], [(1,3),(5,7),(9,11),(13,16),(14,15),(17,20),(18,19),(21,24),(22,23)]])
G:=TransitiveGroup(24,350);
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 6E | ··· | 6L | 6M | 6N | 8A | 8B | 12A | 12B | 12C | 12D | 12E | ··· | 12L | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 8 | 8 | 8 | 8 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | D4 | D4 | D4 | C3xD4 | C3xD4 | C3xD4 | D4:4D4 | C3xD4:4D4 |
| kernel | C3xD4:4D4 | C3xC4.D4 | C3xC4wrC2 | C3xC4:1D4 | C3xC8:C22 | C3x2+ 1+4 | D4:4D4 | C4.D4 | C4wrC2 | C4:1D4 | C8:C22 | 2+ 1+4 | C3xD4 | C3xQ8 | C22xC6 | D4 | Q8 | C23 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 2 | 2 | 4 | 2 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 2 | 4 |
Matrix representation of C3xD4:4D4 ►in GL4(F7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 3 | 2 | 3 | 1 |
| 1 | 0 | 1 | 0 |
| 4 | 4 | 4 | 6 |
| 4 | 3 | 5 | 0 |
| 1 | 4 | 5 | 4 |
| 3 | 4 | 4 | 0 |
| 5 | 2 | 1 | 4 |
| 3 | 3 | 4 | 1 |
| 0 | 1 | 4 | 0 |
| 0 | 1 | 0 | 5 |
| 0 | 0 | 1 | 0 |
| 3 | 4 | 2 | 0 |
| 1 | 0 | 4 | 5 |
| 0 | 1 | 5 | 5 |
| 0 | 0 | 6 | 0 |
| 0 | 0 | 0 | 6 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[3,1,4,4,2,0,4,3,3,1,4,5,1,0,6,0],[1,3,5,3,4,4,2,3,5,4,1,4,4,0,4,1],[0,0,0,3,1,1,0,4,4,0,1,2,0,5,0,0],[1,0,0,0,0,1,0,0,4,5,6,0,5,5,0,6] >;
C3xD4:4D4 in GAP, Magma, Sage, TeX
C_3\times D_4\rtimes_4D_4
% in TeX
G:=Group("C3xD4:4D4"); // GroupNames label
G:=SmallGroup(192,886);
// by ID
G=gap.SmallGroup(192,886);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-2,365,1094,4204,2111,1068,172,3036]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^4=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^-1*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations