metabelian, supersoluble, monomial
Aliases: D12⋊16D6, C32⋊52+ 1+4, Q8⋊8S32, (C4×S3)⋊5D6, (C3×Q8)⋊13D6, (S3×D12)⋊12C2, C3⋊4(D4○D12), Q8⋊3S3⋊8S3, D6⋊D6⋊14C2, (C3×C6).27C24, C6.27(S3×C23), (S3×C12)⋊11C22, D6.D6⋊10C2, C12.26D6⋊8C2, C3⋊D12⋊9C22, (C3×D12)⋊18C22, (S3×C6).15C23, (C3×C12).39C23, C12.39(C22×S3), D6.15(C22×S3), C12⋊S3⋊13C22, D6⋊S3⋊16C22, C32⋊2Q8⋊18C22, C3⋊Dic3.42C23, (Q8×C32)⋊12C22, Dic3.23(C22×S3), (C3×Dic3).22C23, C4.39(C2×S32), (C2×S32)⋊6C22, (C4×C3⋊S3)⋊7C22, C2.29(C22×S32), (C3×Q8⋊3S3)⋊10C2, (C2×C3⋊S3).29C23, SmallGroup(288,968)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12⋊16D6
G = < a,b,c,d | a12=b2=c6=d2=1, bab=dad=a-1, cac-1=a5, cbc-1=a10b, dbd=a4b, dcd=c-1 >
Subgroups: 1530 in 352 conjugacy classes, 108 normal (10 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C2×C4, D4, Q8, Q8, C23, C32, Dic3, Dic3, C12, C12, D6, D6, C2×C6, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, Dic6, C4×S3, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, 2+ 1+4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, C2×C3⋊S3, C2×D12, C4○D12, S3×D4, Q8⋊3S3, Q8⋊3S3, C3×C4○D4, D6⋊S3, C3⋊D12, C32⋊2Q8, S3×C12, C3×D12, C4×C3⋊S3, C12⋊S3, Q8×C32, C2×S32, D4○D12, D6.D6, S3×D12, D6⋊D6, C3×Q8⋊3S3, C12.26D6, D12⋊16D6
Quotients: C1, C2, C22, S3, C23, D6, C24, C22×S3, 2+ 1+4, S32, S3×C23, C2×S32, D4○D12, C22×S32, D12⋊16D6
►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 25)(2 36)(3 35)(4 34)(5 33)(6 32)(7 31)(8 30)(9 29)(10 28)(11 27)(12 26)(13 44)(14 43)(15 42)(16 41)(17 40)(18 39)(19 38)(20 37)(21 48)(22 47)(23 46)(24 45)
(1 13 5 21 9 17)(2 18 6 14 10 22)(3 23 7 19 11 15)(4 16 8 24 12 20)(25 46 33 38 29 42)(26 39 34 43 30 47)(27 44 35 48 31 40)(28 37 36 41 32 45)
(1 8)(2 7)(3 6)(4 5)(9 12)(10 11)(13 16)(14 15)(17 24)(18 23)(19 22)(20 21)(25 26)(27 36)(28 35)(29 34)(30 33)(31 32)(37 44)(38 43)(39 42)(40 41)(45 48)(46 47)
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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45), (1,13,5,21,9,17)(2,18,6,14,10,22)(3,23,7,19,11,15)(4,16,8,24,12,20)(25,46,33,38,29,42)(26,39,34,43,30,47)(27,44,35,48,31,40)(28,37,36,41,32,45), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47)>;
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), (1,25)(2,36)(3,35)(4,34)(5,33)(6,32)(7,31)(8,30)(9,29)(10,28)(11,27)(12,26)(13,44)(14,43)(15,42)(16,41)(17,40)(18,39)(19,38)(20,37)(21,48)(22,47)(23,46)(24,45), (1,13,5,21,9,17)(2,18,6,14,10,22)(3,23,7,19,11,15)(4,16,8,24,12,20)(25,46,33,38,29,42)(26,39,34,43,30,47)(27,44,35,48,31,40)(28,37,36,41,32,45), (1,8)(2,7)(3,6)(4,5)(9,12)(10,11)(13,16)(14,15)(17,24)(18,23)(19,22)(20,21)(25,26)(27,36)(28,35)(29,34)(30,33)(31,32)(37,44)(38,43)(39,42)(40,41)(45,48)(46,47) );
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)], [(1,25),(2,36),(3,35),(4,34),(5,33),(6,32),(7,31),(8,30),(9,29),(10,28),(11,27),(12,26),(13,44),(14,43),(15,42),(16,41),(17,40),(18,39),(19,38),(20,37),(21,48),(22,47),(23,46),(24,45)], [(1,13,5,21,9,17),(2,18,6,14,10,22),(3,23,7,19,11,15),(4,16,8,24,12,20),(25,46,33,38,29,42),(26,39,34,43,30,47),(27,44,35,48,31,40),(28,37,36,41,32,45)], [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11),(13,16),(14,15),(17,24),(18,23),(19,22),(20,21),(25,26),(27,36),(28,35),(29,34),(30,33),(31,32),(37,44),(38,43),(39,42),(40,41),(45,48),(46,47)]])
42 conjugacy classes
| class | 1 | 2A | 2B | ··· | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | ··· | 6I | 12A | ··· | 12F | 12G | 12H | 12I | 12J | 12K | 12L | 12M |
| order | 1 | 2 | 2 | ··· | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | ··· | 6 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 6 | ··· | 6 | 18 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 2 | 6 | 6 | 18 | 2 | 2 | 4 | 12 | ··· | 12 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 8 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D6 | D6 | D6 | 2+ 1+4 | S32 | C2×S32 | D4○D12 | D12⋊16D6 |
| kernel | D12⋊16D6 | D6.D6 | S3×D12 | D6⋊D6 | C3×Q8⋊3S3 | C12.26D6 | Q8⋊3S3 | C4×S3 | D12 | C3×Q8 | C32 | Q8 | C4 | C3 | C1 |
| # reps | 1 | 3 | 6 | 3 | 2 | 1 | 2 | 6 | 6 | 2 | 1 | 1 | 3 | 4 | 1 |
Matrix representation of D12⋊16D6 ►in GL6(𝔽13)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 3 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 3 | 7 | 0 | 0 |
| 0 | 0 | 10 | 10 | 0 | 0 |
| 0 | 0 | 0 | 0 | 3 | 7 |
| 0 | 0 | 0 | 0 | 10 | 10 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 7 | 3 | 0 | 0 |
| 0 | 0 | 10 | 6 | 0 | 0 |
| 0 | 0 | 0 | 0 | 6 | 10 |
| 0 | 0 | 0 | 0 | 3 | 7 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,3,10,0,0,0,0,3,6,0,0,0,0,0,0,10,3,0,0,0,0,10,7],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,3,10,0,0,0,0,7,10,0,0,0,0,0,0,3,10,0,0,0,0,7,10],[0,1,0,0,0,0,12,1,0,0,0,0,0,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0,0,12,0,0,0],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,7,10,0,0,0,0,3,6,0,0,0,0,0,0,6,3,0,0,0,0,10,7] >;
D12⋊16D6 in GAP, Magma, Sage, TeX
D_{12}\rtimes_{16}D_6 % in TeX
G:=Group("D12:16D6"); // GroupNames label
G:=SmallGroup(288,968);
// by ID
G=gap.SmallGroup(288,968);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,100,675,185,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=d^2=1,b*a*b=d*a*d=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^10*b,d*b*d=a^4*b,d*c*d=c^-1>;
// generators/relations