direct product, metacyclic, nilpotent (class 3), monomial, 2-elementary
Aliases: C3×C8.C4, C8.1C12, C24.5C4, C12.68D4, M4(2).2C6, (C2×C8).5C6, (C2×C6).2Q8, C4.8(C2×C12), C22.(C3×Q8), C4.19(C3×D4), C6.14(C4⋊C4), C12.45(C2×C4), (C2×C24).11C2, (C3×M4(2)).4C2, (C2×C12).119C22, C2.5(C3×C4⋊C4), (C2×C4).22(C2×C6), SmallGroup(96,58)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.C4
G = < a,b,c | a3=b8=1, c4=b4, ab=ba, ac=ca, cbc-1=b-1 >
Smallest permutation representation of C3×C8.C4
►On 48 points
(1 25 17)(2 26 18)(3 27 19)(4 28 20)(5 29 21)(6 30 22)(7 31 23)(8 32 24)(9 39 44)(10 40 45)(11 33 46)(12 34 47)(13 35 48)(14 36 41)(15 37 42)(16 38 43)
(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 3 47 5 45 7 43)(2 48 4 46 6 44 8 42)(9 32 15 26 13 28 11 30)(10 31 16 25 14 27 12 29)(17 36 19 34 21 40 23 38)(18 35 20 33 22 39 24 37)
G:=sub<Sym(48)| (1,25,17)(2,26,18)(3,27,19)(4,28,20)(5,29,21)(6,30,22)(7,31,23)(8,32,24)(9,39,44)(10,40,45)(11,33,46)(12,34,47)(13,35,48)(14,36,41)(15,37,42)(16,38,43), (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), (1,41,3,47,5,45,7,43)(2,48,4,46,6,44,8,42)(9,32,15,26,13,28,11,30)(10,31,16,25,14,27,12,29)(17,36,19,34,21,40,23,38)(18,35,20,33,22,39,24,37)>;
G:=Group( (1,25,17)(2,26,18)(3,27,19)(4,28,20)(5,29,21)(6,30,22)(7,31,23)(8,32,24)(9,39,44)(10,40,45)(11,33,46)(12,34,47)(13,35,48)(14,36,41)(15,37,42)(16,38,43), (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), (1,41,3,47,5,45,7,43)(2,48,4,46,6,44,8,42)(9,32,15,26,13,28,11,30)(10,31,16,25,14,27,12,29)(17,36,19,34,21,40,23,38)(18,35,20,33,22,39,24,37) );
G=PermutationGroup([[(1,25,17),(2,26,18),(3,27,19),(4,28,20),(5,29,21),(6,30,22),(7,31,23),(8,32,24),(9,39,44),(10,40,45),(11,33,46),(12,34,47),(13,35,48),(14,36,41),(15,37,42),(16,38,43)], [(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)], [(1,41,3,47,5,45,7,43),(2,48,4,46,6,44,8,42),(9,32,15,26,13,28,11,30),(10,31,16,25,14,27,12,29),(17,36,19,34,21,40,23,38),(18,35,20,33,22,39,24,37)]])
C3×C8.C4 is a maximal subgroup of
C24.7Q8 C24.6Q8 D24.C4 C24.8D4 Dic12.C4 M4(2).25D6 D24⋊10C4 D24⋊7C4 C24.18D4 C24.19D4 C24.42D4
42 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H | 24I | ··· | 24P |
| order | 1 | 2 | 2 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | ··· | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | D4 | Q8 | C3×D4 | C3×Q8 | C8.C4 | C3×C8.C4 |
| kernel | C3×C8.C4 | C2×C24 | C3×M4(2) | C8.C4 | C24 | C2×C8 | M4(2) | C8 | C12 | C2×C6 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 1 | 2 | 2 | 4 | 8 |
Matrix representation of C3×C8.C4 ►in GL2(𝔽73) generated by
| 8 | 0 |
| 0 | 8 |
| 51 | 29 |
| 0 | 63 |
| 21 | 40 |
| 54 | 52 |
G:=sub<GL(2,GF(73))| [8,0,0,8],[51,0,29,63],[21,54,40,52] >;
C3×C8.C4 in GAP, Magma, Sage, TeX
C_3\times C_8.C_4
% in TeX
G:=Group("C3xC8.C4"); // GroupNames label
G:=SmallGroup(96,58);
// by ID
G=gap.SmallGroup(96,58);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-2,144,169,79,1443,117,88]);
// Polycyclic
G:=Group<a,b,c|a^3=b^8=1,c^4=b^4,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations
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