direct product, metacyclic, supersoluble, monomial
Aliases: C3×C8.Dic3, C24.1C12, C12.93D12, C62.12Q8, C24.13Dic3, (C3×C24).7C4, (C2×C24).22S3, (C6×C24).10C2, (C2×C24).11C6, C4.18(C3×D12), C12.34(C3×D4), C4.8(C6×Dic3), C8.1(C3×Dic3), C12.43(C2×C12), (C3×C12).136D4, (C2×C12).434D6, (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), (C3×C12).129(C2×C4), (C2×C12).113(C2×C6), (C3×C4.Dic3).5C2, SmallGroup(288,253)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.Dic3
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 [×2], C3, C4 [×2], C22, C6 [×2], C6 [×5], C8 [×2], C8 [×2], C2×C4, C32, C12 [×4], C12 [×2], C2×C6 [×2], C2×C6, C2×C8, M4(2) [×2], C3×C6, C3×C6, C3⋊C8 [×2], C24 [×4], C24 [×4], C2×C12 [×2], C2×C12, C8.C4, C3×C12 [×2], C62, C4.Dic3 [×2], C2×C24 [×2], C2×C24, C3×M4(2) [×2], C3×C3⋊C8 [×2], C3×C24 [×2], C6×C12, C8.Dic3, C3×C8.C4, C3×C4.Dic3 [×2], C6×C24, C3×C8.Dic3
Quotients: C1, C2 [×3], C3, C4 [×2], C22, S3, C6 [×3], C2×C4, D4, Q8, Dic3 [×2], C12 [×2], D6, C2×C6, C4⋊C4, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C8.C4, C3×Dic3 [×2], S3×C6, C4⋊Dic3, C3×C4⋊C4, C3×Dic6, C3×D12, C6×Dic3, C8.Dic3, C3×C8.C4, C3×C4⋊Dic3, C3×C8.Dic3
►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 22 10 19 7 16 4 13)(2 23 11 20 8 17 5 14)(3 24 12 21 9 18 6 15)(25 40 28 43 31 46 34 37)(26 41 29 44 32 47 35 38)(27 42 30 45 33 48 36 39)
(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 29 10 26 7 35 4 32)(2 34 11 31 8 28 5 25)(3 27 12 36 9 33 6 30)(13 44 22 41 19 38 16 47)(14 37 23 46 20 43 17 40)(15 42 24 39 21 48 18 45)
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), (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,40,28,43,31,46,34,37)(26,41,29,44,32,47,35,38)(27,42,30,45,33,48,36,39), (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), (1,29,10,26,7,35,4,32)(2,34,11,31,8,28,5,25)(3,27,12,36,9,33,6,30)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45)>;
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), (1,22,10,19,7,16,4,13)(2,23,11,20,8,17,5,14)(3,24,12,21,9,18,6,15)(25,40,28,43,31,46,34,37)(26,41,29,44,32,47,35,38)(27,42,30,45,33,48,36,39), (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), (1,29,10,26,7,35,4,32)(2,34,11,31,8,28,5,25)(3,27,12,36,9,33,6,30)(13,44,22,41,19,38,16,47)(14,37,23,46,20,43,17,40)(15,42,24,39,21,48,18,45) );
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)], [(1,22,10,19,7,16,4,13),(2,23,11,20,8,17,5,14),(3,24,12,21,9,18,6,15),(25,40,28,43,31,46,34,37),(26,41,29,44,32,47,35,38),(27,42,30,45,33,48,36,39)], [(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)], [(1,29,10,26,7,35,4,32),(2,34,11,31,8,28,5,25),(3,27,12,36,9,33,6,30),(13,44,22,41,19,38,16,47),(14,37,23,46,20,43,17,40),(15,42,24,39,21,48,18,45)])
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 | C8.Dic3 | C3×C8.C4 | C3×C8.Dic3 |
| kernel | C3×C8.Dic3 | C3×C4.Dic3 | C6×C24 | C8.Dic3 | 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×C8.Dic3 ►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×C8.Dic3 in GAP, Magma, Sage, TeX
C_3\times C_8.{\rm Dic}_3 % in TeX
G:=Group("C3xC8.Dic3"); // 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