direct product, metabelian, supersoluble, monomial
Aliases: C4×C3⋊D12, C12⋊8D12, C62.73C23, D6⋊6(C4×S3), C3⋊4(C4×D12), (C3×C12)⋊16D4, C32⋊10(C4×D4), C12⋊7(C3⋊D4), Dic3⋊4(C4×S3), C6.79(C2×D12), D6⋊Dic3⋊39C2, (Dic3×C12)⋊3C2, (C4×Dic3)⋊16S3, (C2×C12).308D6, C6.31(C4○D12), (C2×Dic3).98D6, (C22×S3).65D6, Dic3⋊Dic3⋊39C2, C6.D12⋊24C2, (C6×C12).234C22, C2.4(D6.D6), (C6×Dic3).113C22, (S3×C2×C4)⋊10S3, C2.21(C4×S32), C3⋊1(C4×C3⋊D4), C6.20(S3×C2×C4), (S3×C2×C12)⋊18C2, (C2×C4).141S32, (S3×C6)⋊12(C2×C4), C22.41(C2×S32), C6.15(C2×C3⋊D4), (C3×Dic3)⋊8(C2×C4), C2.1(C2×C3⋊D12), (C3×C6).100(C2×D4), (S3×C2×C6).78C22, (C3×C6).43(C4○D4), (C2×C6).92(C22×S3), (C3×C6).19(C22×C4), (C2×C3⋊D12).10C2, (C22×C3⋊S3).72C22, (C2×C3⋊Dic3).135C22, (C2×C4×C3⋊S3)⋊13C2, (C2×C3⋊S3)⋊8(C2×C4), SmallGroup(288,551)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×C3⋊D12
G = < a,b,c,d | a4=b3=c12=d2=1, ab=ba, ac=ca, ad=da, cbc-1=dbd=b-1, dcd=c-1 >
Subgroups: 818 in 215 conjugacy classes, 66 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C2×C4, C2×C4, D4, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C22×C4, C2×D4, C3×S3, C3⋊S3, C3×C6, C4×S3, D12, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C4×D4, C3×Dic3, C3×Dic3, C3⋊Dic3, C3×C12, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C4×C12, S3×C2×C4, S3×C2×C4, C2×D12, C2×C3⋊D4, C22×C12, C3⋊D12, S3×C12, C6×Dic3, C4×C3⋊S3, C2×C3⋊Dic3, C6×C12, S3×C2×C6, C22×C3⋊S3, C4×D12, C4×C3⋊D4, D6⋊Dic3, C6.D12, Dic3⋊Dic3, Dic3×C12, C2×C3⋊D12, S3×C2×C12, C2×C4×C3⋊S3, C4×C3⋊D12
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, D12, C3⋊D4, C22×S3, C4×D4, S32, S3×C2×C4, C2×D12, C4○D12, C2×C3⋊D4, C3⋊D12, C2×S32, C4×D12, C4×C3⋊D4, D6.D6, C4×S32, C2×C3⋊D12, C4×C3⋊D12
►On 48 points
(1 29 47 23)(2 30 48 24)(3 31 37 13)(4 32 38 14)(5 33 39 15)(6 34 40 16)(7 35 41 17)(8 36 42 18)(9 25 43 19)(10 26 44 20)(11 27 45 21)(12 28 46 22)
(1 5 9)(2 10 6)(3 7 11)(4 12 8)(13 17 21)(14 22 18)(15 19 23)(16 24 20)(25 29 33)(26 34 30)(27 31 35)(28 36 32)(37 41 45)(38 46 42)(39 43 47)(40 48 44)
(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 3)(4 12)(5 11)(6 10)(7 9)(13 23)(14 22)(15 21)(16 20)(17 19)(25 35)(26 34)(27 33)(28 32)(29 31)(37 47)(38 46)(39 45)(40 44)(41 43)
G:=sub<Sym(48)| (1,29,47,23)(2,30,48,24)(3,31,37,13)(4,32,38,14)(5,33,39,15)(6,34,40,16)(7,35,41,17)(8,36,42,18)(9,25,43,19)(10,26,44,20)(11,27,45,21)(12,28,46,22), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20)(25,29,33)(26,34,30)(27,31,35)(28,36,32)(37,41,45)(38,46,42)(39,43,47)(40,48,44), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43)>;
G:=Group( (1,29,47,23)(2,30,48,24)(3,31,37,13)(4,32,38,14)(5,33,39,15)(6,34,40,16)(7,35,41,17)(8,36,42,18)(9,25,43,19)(10,26,44,20)(11,27,45,21)(12,28,46,22), (1,5,9)(2,10,6)(3,7,11)(4,12,8)(13,17,21)(14,22,18)(15,19,23)(16,24,20)(25,29,33)(26,34,30)(27,31,35)(28,36,32)(37,41,45)(38,46,42)(39,43,47)(40,48,44), (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), (1,3)(4,12)(5,11)(6,10)(7,9)(13,23)(14,22)(15,21)(16,20)(17,19)(25,35)(26,34)(27,33)(28,32)(29,31)(37,47)(38,46)(39,45)(40,44)(41,43) );
G=PermutationGroup([[(1,29,47,23),(2,30,48,24),(3,31,37,13),(4,32,38,14),(5,33,39,15),(6,34,40,16),(7,35,41,17),(8,36,42,18),(9,25,43,19),(10,26,44,20),(11,27,45,21),(12,28,46,22)], [(1,5,9),(2,10,6),(3,7,11),(4,12,8),(13,17,21),(14,22,18),(15,19,23),(16,24,20),(25,29,33),(26,34,30),(27,31,35),(28,36,32),(37,41,45),(38,46,42),(39,43,47),(40,48,44)], [(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)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,23),(14,22),(15,21),(16,20),(17,19),(25,35),(26,34),(27,33),(28,32),(29,31),(37,47),(38,46),(39,45),(40,44),(41,43)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | ··· | 4J | 4K | 4L | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 12A | ··· | 12H | 12I | 12J | 12K | 12L | 12M | ··· | 12X |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 1 | 1 | 1 | 1 | 6 | ··· | 6 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D6 | D6 | D6 | C4○D4 | C4×S3 | D12 | C3⋊D4 | C4×S3 | C4○D12 | S32 | C3⋊D12 | C2×S32 | D6.D6 | C4×S32 |
| kernel | C4×C3⋊D12 | D6⋊Dic3 | C6.D12 | Dic3⋊Dic3 | Dic3×C12 | C2×C3⋊D12 | S3×C2×C12 | C2×C4×C3⋊S3 | C3⋊D12 | C4×Dic3 | S3×C2×C4 | C3×C12 | C2×Dic3 | C2×C12 | C22×S3 | C3×C6 | Dic3 | C12 | C12 | D6 | C6 | C2×C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 2 | 3 | 2 | 1 | 2 | 4 | 4 | 4 | 4 | 8 | 1 | 2 | 1 | 2 | 2 |
Matrix representation of C4×C3⋊D12 ►in GL6(𝔽13)
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 12 | 12 |
G:=sub<GL(6,GF(13))| [12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,0,5],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,1,0,0,0,0,12,0],[12,1,0,0,0,0,12,0,0,0,0,0,0,0,0,12,0,0,0,0,1,0,0,0,0,0,0,0,12,1,0,0,0,0,0,1],[12,0,0,0,0,0,12,1,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,12,0,0,0,0,0,12] >;
C4×C3⋊D12 in GAP, Magma, Sage, TeX
C_4\times C_3\rtimes D_{12} % in TeX
G:=Group("C4xC3:D12"); // GroupNames label
G:=SmallGroup(288,551);
// by ID
G=gap.SmallGroup(288,551);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,120,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^12=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