direct product, metabelian, supersoluble, monomial
Aliases: C3×C12.48D4, C62⋊9Q8, C62.192C23, C6.7(C6×Q8), C4⋊Dic3⋊8C6, (C2×C6)⋊9Dic6, C6.39(C6×D4), Dic3⋊C4⋊2C6, (C2×Dic6)⋊6C6, C12.57(C3×D4), C2.9(C6×Dic6), (C6×Dic6)⋊30C2, (C2×C12).351D6, (C3×C12).159D4, C23.30(S3×C6), C6.54(C2×Dic6), C22⋊4(C3×Dic6), (C22×C12).40S3, (C22×C12).19C6, C6.D4.4C6, (C22×C6).125D6, C6.124(C4○D12), C32⋊22(C22⋊Q8), C12.140(C3⋊D4), (C6×C12).324C22, (C2×C62).95C22, (C6×Dic3).97C22, (C2×C6)⋊5(C3×Q8), (C2×C6×C12).15C2, C2.5(C6×C3⋊D4), C3⋊4(C3×C22⋊Q8), (C2×C4).83(S3×C6), C6.14(C3×C4○D4), C4.23(C3×C3⋊D4), C22.54(S3×C2×C6), (C3×C6).50(C2×Q8), (C2×C12).91(C2×C6), (C3×C4⋊Dic3)⋊32C2, C2.17(C3×C4○D12), (C3×C6).251(C2×D4), C6.140(C2×C3⋊D4), (C22×C4).9(C3×S3), (C3×Dic3⋊C4)⋊34C2, (C2×C6).47(C22×C6), (C22×C6).59(C2×C6), (C2×Dic3).9(C2×C6), (C3×C6).102(C4○D4), (C2×C6).325(C22×S3), (C3×C6.D4).9C2, SmallGroup(288,695)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C12.48D4
G = < a,b,c,d | a3=b12=c4=1, d2=b6, ab=ba, ac=ca, ad=da, cbc-1=dbd-1=b-1, dcd-1=b6c-1 >
Subgroups: 346 in 175 conjugacy classes, 74 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, C6, C6, C2×C4, C2×C4, Q8, C23, C32, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C22×C4, C2×Q8, C3×C6, C3×C6, Dic6, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×C6, C22×C6, C22⋊Q8, C3×Dic3, C3×C12, C3×C12, C62, C62, C62, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C22×C12, C6×Q8, C3×Dic6, C6×Dic3, C6×C12, C6×C12, C2×C62, C12.48D4, C3×C22⋊Q8, C3×Dic3⋊C4, C3×C4⋊Dic3, C3×C6.D4, C6×Dic6, C2×C6×C12, C3×C12.48D4
Quotients: C1, C2, C3, C22, S3, C6, D4, Q8, C23, D6, C2×C6, C2×D4, C2×Q8, C4○D4, C3×S3, Dic6, C3⋊D4, C3×D4, C3×Q8, C22×S3, C22×C6, C22⋊Q8, S3×C6, C2×Dic6, C4○D12, C2×C3⋊D4, C6×D4, C6×Q8, C3×C4○D4, C3×Dic6, C3×C3⋊D4, S3×C2×C6, C12.48D4, C3×C22⋊Q8, C6×Dic6, C3×C4○D12, C6×C3⋊D4, C3×C12.48D4
►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 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 28 14 43)(2 27 15 42)(3 26 16 41)(4 25 17 40)(5 36 18 39)(6 35 19 38)(7 34 20 37)(8 33 21 48)(9 32 22 47)(10 31 23 46)(11 30 24 45)(12 29 13 44)
(1 43 7 37)(2 42 8 48)(3 41 9 47)(4 40 10 46)(5 39 11 45)(6 38 12 44)(13 29 19 35)(14 28 20 34)(15 27 21 33)(16 26 22 32)(17 25 23 31)(18 36 24 30)
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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,28,14,43)(2,27,15,42)(3,26,16,41)(4,25,17,40)(5,36,18,39)(6,35,19,38)(7,34,20,37)(8,33,21,48)(9,32,22,47)(10,31,23,46)(11,30,24,45)(12,29,13,44), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30)>;
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), (1,2,3,4,5,6,7,8,9,10,11,12)(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), (1,28,14,43)(2,27,15,42)(3,26,16,41)(4,25,17,40)(5,36,18,39)(6,35,19,38)(7,34,20,37)(8,33,21,48)(9,32,22,47)(10,31,23,46)(11,30,24,45)(12,29,13,44), (1,43,7,37)(2,42,8,48)(3,41,9,47)(4,40,10,46)(5,39,11,45)(6,38,12,44)(13,29,19,35)(14,28,20,34)(15,27,21,33)(16,26,22,32)(17,25,23,31)(18,36,24,30) );
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)], [(1,2,3,4,5,6,7,8,9,10,11,12),(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)], [(1,28,14,43),(2,27,15,42),(3,26,16,41),(4,25,17,40),(5,36,18,39),(6,35,19,38),(7,34,20,37),(8,33,21,48),(9,32,22,47),(10,31,23,46),(11,30,24,45),(12,29,13,44)], [(1,43,7,37),(2,42,8,48),(3,41,9,47),(4,40,10,46),(5,39,11,45),(6,38,12,44),(13,29,19,35),(14,28,20,34),(15,27,21,33),(16,26,22,32),(17,25,23,31),(18,36,24,30)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6AE | 12A | ··· | 12AF | 12AG | ··· | 12AN |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | ··· | 2 | 12 | ··· | 12 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 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 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | Q8 | D6 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | Dic6 | C3×Q8 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C3⋊D4 | C3×Dic6 | C3×C4○D12 |
| kernel | C3×C12.48D4 | C3×Dic3⋊C4 | C3×C4⋊Dic3 | C3×C6.D4 | C6×Dic6 | C2×C6×C12 | C12.48D4 | Dic3⋊C4 | C4⋊Dic3 | C6.D4 | C2×Dic6 | C22×C12 | C22×C12 | C3×C12 | C62 | C2×C12 | C22×C6 | C3×C6 | C22×C4 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C23 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 1 | 2 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C3×C12.48D4 ►in GL4(𝔽13) generated by
| 3 | 0 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 0 | 9 |
| 6 | 0 | 0 | 0 |
| 4 | 11 | 0 | 0 |
| 0 | 0 | 11 | 0 |
| 0 | 0 | 0 | 6 |
| 1 | 11 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 12 |
| 0 | 0 | 1 | 0 |
| 12 | 2 | 0 | 0 |
| 12 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 12 | 0 |
G:=sub<GL(4,GF(13))| [3,0,0,0,0,3,0,0,0,0,9,0,0,0,0,9],[6,4,0,0,0,11,0,0,0,0,11,0,0,0,0,6],[1,0,0,0,11,12,0,0,0,0,0,1,0,0,12,0],[12,12,0,0,2,1,0,0,0,0,0,12,0,0,1,0] >;
C3×C12.48D4 in GAP, Magma, Sage, TeX
C_3\times C_{12}._{48}D_4 % in TeX
G:=Group("C3xC12.48D4"); // GroupNames label
G:=SmallGroup(288,695);
// by ID
G=gap.SmallGroup(288,695);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,336,701,344,590,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^12=c^4=1,d^2=b^6,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d^-1=b^-1,d*c*d^-1=b^6*c^-1>;
// generators/relations