direct product, metacyclic, supersoluble, monomial
Aliases: C3×C24.C4, C24.1C12, C12.93D12, C62.12Q8, C24.13Dic3, (C3×C24).7C4, (C6×C24).10C2, (C2×C24).22S3, (C2×C24).11C6, C12.34(C3×D4), C4.18(C3×D12), C4.8(C6×Dic3), C8.1(C3×Dic3), C12.43(C2×C12), (C2×C12).434D6, (C3×C12).136D4, (C2×C6).11Dic6, C4.Dic3.1C6, C6.20(C4⋊Dic3), C32⋊7(C8.C4), C12.65(C2×Dic3), C22.2(C3×Dic6), (C6×C12).312C22, C6.7(C3×C4⋊C4), (C2×C8).5(C3×S3), (C2×C6).6(C3×Q8), C3⋊1(C3×C8.C4), (C2×C4).73(S3×C6), C2.5(C3×C4⋊Dic3), (C3×C6).33(C4⋊C4), (C2×C12).113(C2×C6), (C3×C12).129(C2×C4), (C3×C4.Dic3).5C2, SmallGroup(288,253)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24.C4
G = < a,b,c,d | a3=b8=1, c6=b4, d2=b4c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b-1, dcd-1=c5 >
Subgroups: 122 in 71 conjugacy classes, 42 normal (38 characteristic)
C1, C2, C2, C3, C3, C4, C22, C6, C6, C8, C8, C2×C4, C32, C12, C12, C2×C6, C2×C6, C2×C8, M4(2), C3×C6, C3×C6, C3⋊C8, C24, C24, C2×C12, C2×C12, C8.C4, C3×C12, C62, C4.Dic3, C2×C24, C2×C24, C3×M4(2), C3×C3⋊C8, C3×C24, C6×C12, C24.C4, C3×C8.C4, C3×C4.Dic3, C6×C24, C3×C24.C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C4, C3×Dic3, S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C24.C4, C3×C8.C4, C3×C4⋊Dic3, C3×C24.C4
►On 48 points
(1 9 5)(2 10 6)(3 11 7)(4 12 8)(13 21 17)(14 22 18)(15 23 19)(16 24 20)(25 29 33)(26 30 34)(27 31 35)(28 32 36)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 19 10 16 7 13 4 22)(2 20 11 17 8 14 5 23)(3 21 12 18 9 15 6 24)(25 43 28 46 31 37 34 40)(26 44 29 47 32 38 35 41)(27 45 30 48 33 39 36 42)
(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 33 10 30 7 27 4 36)(2 26 11 35 8 32 5 29)(3 31 12 28 9 25 6 34)(13 42 22 39 19 48 16 45)(14 47 23 44 20 41 17 38)(15 40 24 37 21 46 18 43)
G:=sub<Sym(48)| (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,19,10,16,7,13,4,22)(2,20,11,17,8,14,5,23)(3,21,12,18,9,15,6,24)(25,43,28,46,31,37,34,40)(26,44,29,47,32,38,35,41)(27,45,30,48,33,39,36,42), (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), (1,33,10,30,7,27,4,36)(2,26,11,35,8,32,5,29)(3,31,12,28,9,25,6,34)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43)>;
G:=Group( (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,21,17)(14,22,18)(15,23,19)(16,24,20)(25,29,33)(26,30,34)(27,31,35)(28,32,36)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,19,10,16,7,13,4,22)(2,20,11,17,8,14,5,23)(3,21,12,18,9,15,6,24)(25,43,28,46,31,37,34,40)(26,44,29,47,32,38,35,41)(27,45,30,48,33,39,36,42), (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), (1,33,10,30,7,27,4,36)(2,26,11,35,8,32,5,29)(3,31,12,28,9,25,6,34)(13,42,22,39,19,48,16,45)(14,47,23,44,20,41,17,38)(15,40,24,37,21,46,18,43) );
G=PermutationGroup([[(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,21,17),(14,22,18),(15,23,19),(16,24,20),(25,29,33),(26,30,34),(27,31,35),(28,32,36),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,19,10,16,7,13,4,22),(2,20,11,17,8,14,5,23),(3,21,12,18,9,15,6,24),(25,43,28,46,31,37,34,40),(26,44,29,47,32,38,35,41),(27,45,30,48,33,39,36,42)], [(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)], [(1,33,10,30,7,27,4,36),(2,26,11,35,8,32,5,29),(3,31,12,28,9,25,6,34),(13,42,22,39,19,48,16,45),(14,47,23,44,20,41,17,38),(15,40,24,37,21,46,18,43)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | 6B | 6C | ··· | 6M | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12R | 24A | ··· | 24AF | 24AG | ··· | 24AN |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 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 | 2 | 2 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | - | - | + | + | - | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | D4 | Q8 | Dic3 | D6 | C3×S3 | D12 | C3×D4 | Dic6 | C3×Q8 | C8.C4 | C3×Dic3 | S3×C6 | C3×D12 | C3×Dic6 | C24.C4 | C3×C8.C4 | C3×C24.C4 |
| kernel | C3×C24.C4 | C3×C4.Dic3 | C6×C24 | C24.C4 | C3×C24 | C4.Dic3 | C2×C24 | C24 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C32 | C8 | C2×C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of C3×C24.C4 ►in GL4(𝔽5) generated by
| 3 | 0 | 4 | 0 |
| 0 | 1 | 0 | 2 |
| 3 | 0 | 1 | 0 |
| 0 | 1 | 0 | 3 |
| 3 | 0 | 2 | 0 |
| 0 | 1 | 0 | 3 |
| 4 | 0 | 2 | 0 |
| 0 | 4 | 0 | 4 |
| 3 | 0 | 3 | 0 |
| 0 | 3 | 0 | 1 |
| 1 | 0 | 4 | 0 |
| 0 | 3 | 0 | 4 |
| 0 | 4 | 0 | 4 |
| 3 | 0 | 4 | 0 |
| 0 | 4 | 0 | 2 |
| 4 | 0 | 1 | 0 |
G:=sub<GL(4,GF(5))| [3,0,3,0,0,1,0,1,4,0,1,0,0,2,0,3],[3,0,4,0,0,1,0,4,2,0,2,0,0,3,0,4],[3,0,1,0,0,3,0,3,3,0,4,0,0,1,0,4],[0,3,0,4,4,0,4,0,0,4,0,1,4,0,2,0] >;
C3×C24.C4 in GAP, Magma, Sage, TeX
C_3\times C_{24}.C_4 % in TeX
G:=Group("C3xC24.C4"); // GroupNames label
G:=SmallGroup(288,253);
// by ID
G=gap.SmallGroup(288,253);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,136,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^4,d^2=b^4*c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^-1,d*c*d^-1=c^5>;
// generators/relations