direct product, metabelian, supersoluble, monomial
Aliases: C12×C3⋊D4, C62.196C23, C3⋊4(D4×C12), C12⋊8(C3×D4), D6⋊C4⋊18C6, D6⋊4(C2×C12), (C3×C12)⋊24D4, C6.41(C6×D4), C32⋊22(C4×D4), C22⋊4(S3×C12), C62⋊18(C2×C4), (C22×C12)⋊6S3, Dic3⋊C4⋊18C6, (C22×C12)⋊13C6, (C4×Dic3)⋊16C6, Dic3⋊2(C2×C12), (C2×C12).352D6, C23.32(S3×C6), (Dic3×C12)⋊34C2, C6.D4⋊14C6, C6.19(C22×C12), (C22×C6).127D6, C6.126(C4○D12), (C6×C12).352C22, (C2×C62).99C22, (C6×Dic3).135C22, (S3×C2×C4)⋊14C6, (C2×C6×C12)⋊14C2, (S3×C2×C12)⋊28C2, (C2×C6)⋊13(C4×S3), (C2×C6)⋊8(C2×C12), C2.20(S3×C2×C12), C6.119(S3×C2×C4), C2.3(C6×C3⋊D4), (S3×C6)⋊17(C2×C4), (C3×D6⋊C4)⋊39C2, (C22×C4)⋊6(C3×S3), C6.16(C3×C4○D4), C2.5(C3×C4○D12), (C2×C3⋊D4).7C6, C22.24(S3×C2×C6), (C2×C4).103(S3×C6), (C3×C6).252(C2×D4), (C6×C3⋊D4).14C2, C6.142(C2×C3⋊D4), (S3×C2×C6).98C22, (C2×C12).132(C2×C6), (C3×Dic3⋊C4)⋊40C2, (C3×Dic3)⋊11(C2×C4), (C2×C6).51(C22×C6), (C3×C6).91(C22×C4), (C22×C6).63(C2×C6), (C3×C6).104(C4○D4), (C3×C6.D4)⋊30C2, (C22×S3).25(C2×C6), (C2×C6).329(C22×S3), (C2×Dic3).35(C2×C6), SmallGroup(288,699)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12×C3⋊D4
G = < a,b,c,d | a12=b3=c4=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 442 in 215 conjugacy classes, 90 normal (58 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, C62, C62, C62, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C4×C12, C3×C22⋊C4, C3×C4⋊C4, S3×C2×C4, C2×C3⋊D4, C22×C12, C22×C12, C6×D4, S3×C12, C6×Dic3, C3×C3⋊D4, C6×C12, C6×C12, S3×C2×C6, C2×C62, C4×C3⋊D4, D4×C12, Dic3×C12, C3×Dic3⋊C4, C3×D6⋊C4, C3×C6.D4, S3×C2×C12, C6×C3⋊D4, C2×C6×C12, C12×C3⋊D4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C23, C12, D6, C2×C6, C22×C4, C2×D4, C4○D4, C3×S3, C4×S3, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×C6, C4×D4, S3×C6, S3×C2×C4, C4○D12, C2×C3⋊D4, C22×C12, C6×D4, C3×C4○D4, S3×C12, C3×C3⋊D4, S3×C2×C6, C4×C3⋊D4, D4×C12, S3×C2×C12, C3×C4○D12, C6×C3⋊D4, C12×C3⋊D4
►On 48 points
(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 9 5)(2 10 6)(3 11 7)(4 12 8)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 41 45)(38 42 46)(39 43 47)(40 44 48)
(1 19 29 48)(2 20 30 37)(3 21 31 38)(4 22 32 39)(5 23 33 40)(6 24 34 41)(7 13 35 42)(8 14 36 43)(9 15 25 44)(10 16 26 45)(11 17 27 46)(12 18 28 47)
(1 42)(2 43)(3 44)(4 45)(5 46)(6 47)(7 48)(8 37)(9 38)(10 39)(11 40)(12 41)(13 29)(14 30)(15 31)(16 32)(17 33)(18 34)(19 35)(20 36)(21 25)(22 26)(23 27)(24 28)
G:=sub<Sym(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), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,19,29,48)(2,20,30,37)(3,21,31,38)(4,22,32,39)(5,23,33,40)(6,24,34,41)(7,13,35,42)(8,14,36,43)(9,15,25,44)(10,16,26,45)(11,17,27,46)(12,18,28,47), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28)>;
G:=Group( (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), (1,9,5)(2,10,6)(3,11,7)(4,12,8)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,41,45)(38,42,46)(39,43,47)(40,44,48), (1,19,29,48)(2,20,30,37)(3,21,31,38)(4,22,32,39)(5,23,33,40)(6,24,34,41)(7,13,35,42)(8,14,36,43)(9,15,25,44)(10,16,26,45)(11,17,27,46)(12,18,28,47), (1,42)(2,43)(3,44)(4,45)(5,46)(6,47)(7,48)(8,37)(9,38)(10,39)(11,40)(12,41)(13,29)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(20,36)(21,25)(22,26)(23,27)(24,28) );
G=PermutationGroup([[(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)], [(1,9,5),(2,10,6),(3,11,7),(4,12,8),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,41,45),(38,42,46),(39,43,47),(40,44,48)], [(1,19,29,48),(2,20,30,37),(3,21,31,38),(4,22,32,39),(5,23,33,40),(6,24,34,41),(7,13,35,42),(8,14,36,43),(9,15,25,44),(10,16,26,45),(11,17,27,46),(12,18,28,47)], [(1,42),(2,43),(3,44),(4,45),(5,46),(6,47),(7,48),(8,37),(9,38),(10,39),(11,40),(12,41),(13,29),(14,30),(15,31),(16,32),(17,33),(18,34),(19,35),(20,36),(21,25),(22,26),(23,27),(24,28)]])
108 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4L | 6A | ··· | 6F | 6G | ··· | 6AE | 6AF | 6AG | 6AH | 6AI | 12A | ··· | 12H | 12I | ··· | 12AJ | 12AK | ··· | 12AV |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | 6 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 6 | ··· | 6 |
108 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 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 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C6 | C6 | C6 | C12 | S3 | D4 | D6 | D6 | C4○D4 | C3×S3 | C3⋊D4 | C3×D4 | C4×S3 | S3×C6 | S3×C6 | C4○D12 | C3×C4○D4 | C3×C3⋊D4 | S3×C12 | C3×C4○D12 |
| kernel | C12×C3⋊D4 | Dic3×C12 | C3×Dic3⋊C4 | C3×D6⋊C4 | C3×C6.D4 | S3×C2×C12 | C6×C3⋊D4 | C2×C6×C12 | C4×C3⋊D4 | C3×C3⋊D4 | C4×Dic3 | Dic3⋊C4 | D6⋊C4 | C6.D4 | S3×C2×C4 | C2×C3⋊D4 | C22×C12 | C3⋊D4 | C22×C12 | C3×C12 | C2×C12 | C22×C6 | C3×C6 | C22×C4 | C12 | C12 | C2×C6 | C2×C4 | C23 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 8 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 16 | 1 | 2 | 2 | 1 | 2 | 2 | 4 | 4 | 4 | 4 | 2 | 4 | 4 | 8 | 8 | 8 |
Matrix representation of C12×C3⋊D4 ►in GL3(𝔽13) generated by
| 5 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 7 |
| 1 | 0 | 0 |
| 0 | 3 | 0 |
| 0 | 9 | 9 |
| 1 | 0 | 0 |
| 0 | 1 | 5 |
| 0 | 10 | 12 |
| 1 | 0 | 0 |
| 0 | 12 | 8 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(13))| [5,0,0,0,7,0,0,0,7],[1,0,0,0,3,9,0,0,9],[1,0,0,0,1,10,0,5,12],[1,0,0,0,12,0,0,8,1] >;
C12×C3⋊D4 in GAP, Magma, Sage, TeX
C_{12}\times C_3\rtimes D_4 % in TeX
G:=Group("C12xC3:D4"); // GroupNames label
G:=SmallGroup(288,699);
// by ID
G=gap.SmallGroup(288,699);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,142,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^3=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=b^-1,d*c*d=c^-1>;
// generators/relations