direct product, metabelian, nilpotent (class 2), monomial, 2-elementary
Aliases: C3×C4.10C42, C12.33C42, (C2×C24).4C4, (C2×C8).1C12, C4.10(C4×C12), (C2×C12).278D4, (C22×C6).1Q8, C23.1(C3×Q8), (C2×M4(2)).5C6, (C6×M4(2)).17C2, C12.102(C22⋊C4), C6.22(C2.C42), (C22×C12).387C22, (C2×C4).9(C3×D4), C22.2(C3×C4⋊C4), (C2×C6).19(C4⋊C4), (C2×C4).64(C2×C12), C4.18(C3×C22⋊C4), (C2×C12).325(C2×C4), (C22×C4).22(C2×C6), C2.3(C3×C2.C42), SmallGroup(192,144)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C4.10C42
G = < a,b,c,d | a3=b4=1, c4=d4=b2, ab=ba, ac=ca, ad=da, dcd-1=bc=cb, bd=db >
Subgroups: 122 in 86 conjugacy classes, 54 normal (12 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C23, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C22×C4, C24, C2×C12, C22×C6, C2×M4(2), C2×C24, C3×M4(2), C22×C12, C4.10C42, C6×M4(2), C3×C4.10C42
Quotients: C1, C2, C3, C4, C22, C6, C2×C4, D4, Q8, C12, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C12, C3×D4, C3×Q8, C2.C42, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C4.10C42, C3×C2.C42, C3×C4.10C42
►On 48 points
(1 39 19)(2 40 20)(3 33 21)(4 34 22)(5 35 23)(6 36 24)(7 37 17)(8 38 18)(9 46 25)(10 47 26)(11 48 27)(12 41 28)(13 42 29)(14 43 30)(15 44 31)(16 45 32)
(1 3 5 7)(2 4 6 8)(9 15 13 11)(10 16 14 12)(17 19 21 23)(18 20 22 24)(25 31 29 27)(26 32 30 28)(33 35 37 39)(34 36 38 40)(41 47 45 43)(42 48 46 44)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)
(1 41 7 43 5 45 3 47)(2 44 4 42 6 48 8 46)(9 20 15 22 13 24 11 18)(10 19 12 17 14 23 16 21)(25 40 31 34 29 36 27 38)(26 39 28 37 30 35 32 33)
G:=sub<Sym(48)| (1,39,19)(2,40,20)(3,33,21)(4,34,22)(5,35,23)(6,36,24)(7,37,17)(8,38,18)(9,46,25)(10,47,26)(11,48,27)(12,41,28)(13,42,29)(14,43,30)(15,44,31)(16,45,32), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,41,7,43,5,45,3,47)(2,44,4,42,6,48,8,46)(9,20,15,22,13,24,11,18)(10,19,12,17,14,23,16,21)(25,40,31,34,29,36,27,38)(26,39,28,37,30,35,32,33)>;
G:=Group( (1,39,19)(2,40,20)(3,33,21)(4,34,22)(5,35,23)(6,36,24)(7,37,17)(8,38,18)(9,46,25)(10,47,26)(11,48,27)(12,41,28)(13,42,29)(14,43,30)(15,44,31)(16,45,32), (1,3,5,7)(2,4,6,8)(9,15,13,11)(10,16,14,12)(17,19,21,23)(18,20,22,24)(25,31,29,27)(26,32,30,28)(33,35,37,39)(34,36,38,40)(41,47,45,43)(42,48,46,44), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48), (1,41,7,43,5,45,3,47)(2,44,4,42,6,48,8,46)(9,20,15,22,13,24,11,18)(10,19,12,17,14,23,16,21)(25,40,31,34,29,36,27,38)(26,39,28,37,30,35,32,33) );
G=PermutationGroup([[(1,39,19),(2,40,20),(3,33,21),(4,34,22),(5,35,23),(6,36,24),(7,37,17),(8,38,18),(9,46,25),(10,47,26),(11,48,27),(12,41,28),(13,42,29),(14,43,30),(15,44,31),(16,45,32)], [(1,3,5,7),(2,4,6,8),(9,15,13,11),(10,16,14,12),(17,19,21,23),(18,20,22,24),(25,31,29,27),(26,32,30,28),(33,35,37,39),(34,36,38,40),(41,47,45,43),(42,48,46,44)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48)], [(1,41,7,43,5,45,3,47),(2,44,4,42,6,48,8,46),(9,20,15,22,13,24,11,18),(10,19,12,17,14,23,16,21),(25,40,31,34,29,36,27,38),(26,39,28,37,30,35,32,33)]])
66 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | ··· | 6H | 8A | ··· | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 24A | ··· | 24X |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 4 | ··· | 4 |
66 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | - | ||||||||
| image | C1 | C2 | C3 | C4 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C4.10C42 | C3×C4.10C42 |
| kernel | C3×C4.10C42 | C6×M4(2) | C4.10C42 | C2×C24 | C2×M4(2) | C2×C8 | C2×C12 | C22×C6 | C2×C4 | C23 | C3 | C1 |
| # reps | 1 | 3 | 2 | 12 | 6 | 24 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of C3×C4.10C42 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 0 | 64 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 46 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 |
| 0 | 0 | 1 | 0 |
G:=sub<GL(4,GF(73))| [64,0,0,0,0,64,0,0,0,0,64,0,0,0,0,64],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,0,46],[0,0,46,0,0,0,0,27,1,0,0,0,0,1,0,0],[0,27,0,0,1,0,0,0,0,0,0,1,0,0,46,0] >;
C3×C4.10C42 in GAP, Magma, Sage, TeX
C_3\times C_4._{10}C_4^2 % in TeX
G:=Group("C3xC4.10C4^2"); // GroupNames label
G:=SmallGroup(192,144);
// by ID
G=gap.SmallGroup(192,144);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-2,168,197,344,248,2111,172,6053,124]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^4=1,c^4=d^4=b^2,a*b=b*a,a*c=c*a,a*d=d*a,d*c*d^-1=b*c=c*b,b*d=d*b>;
// generators/relations