metabelian, supersoluble, monomial
Aliases: D12.28D6, C12.23D12, C62.49D4, C3⋊C8⋊4D6, (C2×D12)⋊9S3, (C6×D12)⋊4C2, C3⋊D24⋊6C2, C3⋊4(C8⋊D6), (C3×C12).67D4, C6.69(C2×D12), (C2×C6).60D12, C4.Dic3⋊8S3, (C2×C12).119D6, C32⋊8(C8⋊C22), D12.S3⋊3C2, C12.59D6⋊3C2, C3⋊1(D12⋊6C22), C12.47(C3⋊D4), (C6×C12).79C22, (C3×C12).66C23, C12⋊S3⋊17C22, C4.23(C3⋊D12), C12.126(C22×S3), (C3×D12).36C22, C32⋊4Q8⋊16C22, C22.4(C3⋊D12), (C2×C4).8S32, C4.54(C2×S32), (C3×C3⋊C8)⋊4C22, C6.5(C2×C3⋊D4), (C3×C6).70(C2×D4), C2.9(C2×C3⋊D12), (C3×C4.Dic3)⋊2C2, (C2×C6).19(C3⋊D4), SmallGroup(288,478)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D12.28D6
G = < a,b,c,d | a12=b2=c6=1, d2=a3, bab=a-1, ac=ca, ad=da, bc=cb, dbd-1=a3b, dcd-1=a6c-1 >
Subgroups: 698 in 156 conjugacy classes, 44 normal (32 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, M4(2), D8, SD16, C2×D4, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3⋊C8, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C8⋊C22, C3⋊Dic3, C3×C12, S3×C6, C2×C3⋊S3, C62, C24⋊C2, D24, C4.Dic3, D4⋊S3, D4.S3, C3×M4(2), C2×D12, C4○D12, C6×D4, C3×C3⋊C8, C3×D12, C3×D12, C32⋊4Q8, C4×C3⋊S3, C12⋊S3, C32⋊7D4, C6×C12, S3×C2×C6, C8⋊D6, D12⋊6C22, C3⋊D24, D12.S3, C3×C4.Dic3, C6×D12, C12.59D6, D12.28D6
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C3⋊D4, C22×S3, C8⋊C22, S32, C2×D12, C2×C3⋊D4, C3⋊D12, C2×S32, C8⋊D6, D12⋊6C22, C2×C3⋊D12, D12.28D6
►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 39)(2 38)(3 37)(4 48)(5 47)(6 46)(7 45)(8 44)(9 43)(10 42)(11 41)(12 40)(13 27)(14 26)(15 25)(16 36)(17 35)(18 34)(19 33)(20 32)(21 31)(22 30)(23 29)(24 28)
(1 18 5 22 9 14)(2 19 6 23 10 15)(3 20 7 24 11 16)(4 21 8 13 12 17)(25 38 33 46 29 42)(26 39 34 47 30 43)(27 40 35 48 31 44)(28 41 36 37 32 45)
(1 39 4 42 7 45 10 48)(2 40 5 43 8 46 11 37)(3 41 6 44 9 47 12 38)(13 27 16 30 19 33 22 36)(14 28 17 31 20 34 23 25)(15 29 18 32 21 35 24 26)
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,18,5,22,9,14)(2,19,6,23,10,15)(3,20,7,24,11,16)(4,21,8,13,12,17)(25,38,33,46,29,42)(26,39,34,47,30,43)(27,40,35,48,31,44)(28,41,36,37,32,45), (1,39,4,42,7,45,10,48)(2,40,5,43,8,46,11,37)(3,41,6,44,9,47,12,38)(13,27,16,30,19,33,22,36)(14,28,17,31,20,34,23,25)(15,29,18,32,21,35,24,26)>;
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), (1,39)(2,38)(3,37)(4,48)(5,47)(6,46)(7,45)(8,44)(9,43)(10,42)(11,41)(12,40)(13,27)(14,26)(15,25)(16,36)(17,35)(18,34)(19,33)(20,32)(21,31)(22,30)(23,29)(24,28), (1,18,5,22,9,14)(2,19,6,23,10,15)(3,20,7,24,11,16)(4,21,8,13,12,17)(25,38,33,46,29,42)(26,39,34,47,30,43)(27,40,35,48,31,44)(28,41,36,37,32,45), (1,39,4,42,7,45,10,48)(2,40,5,43,8,46,11,37)(3,41,6,44,9,47,12,38)(13,27,16,30,19,33,22,36)(14,28,17,31,20,34,23,25)(15,29,18,32,21,35,24,26) );
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)], [(1,39),(2,38),(3,37),(4,48),(5,47),(6,46),(7,45),(8,44),(9,43),(10,42),(11,41),(12,40),(13,27),(14,26),(15,25),(16,36),(17,35),(18,34),(19,33),(20,32),(21,31),(22,30),(23,29),(24,28)], [(1,18,5,22,9,14),(2,19,6,23,10,15),(3,20,7,24,11,16),(4,21,8,13,12,17),(25,38,33,46,29,42),(26,39,34,47,30,43),(27,40,35,48,31,44),(28,41,36,37,32,45)], [(1,39,4,42,7,45,10,48),(2,40,5,43,8,46,11,37),(3,41,6,44,9,47,12,38),(13,27,16,30,19,33,22,36),(14,28,17,31,20,34,23,25),(15,29,18,32,21,35,24,26)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 8A | 8B | 12A | 12B | 12C | ··· | 12I | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 2 | 12 | 12 | 36 | 2 | 2 | 4 | 2 | 2 | 36 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 12 | 12 | 2 | 2 | 4 | ··· | 4 | 12 | 12 | 12 | 12 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | C3⋊D4 | D12 | C3⋊D4 | C8⋊C22 | S32 | C3⋊D12 | C2×S32 | C3⋊D12 | C8⋊D6 | D12⋊6C22 | D12.28D6 |
| kernel | D12.28D6 | C3⋊D24 | D12.S3 | C3×C4.Dic3 | C6×D12 | C12.59D6 | C4.Dic3 | C2×D12 | C3×C12 | C62 | C3⋊C8 | D12 | C2×C12 | C12 | C12 | C2×C6 | C2×C6 | C32 | C2×C4 | C4 | C4 | C22 | C3 | C3 | C1 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 4 |
Matrix representation of D12.28D6 ►in GL4(𝔽73) generated by
| 7 | 66 | 0 | 0 |
| 7 | 14 | 0 | 0 |
| 0 | 0 | 7 | 66 |
| 0 | 0 | 7 | 14 |
| 0 | 0 | 27 | 27 |
| 0 | 0 | 0 | 46 |
| 46 | 46 | 0 | 0 |
| 0 | 27 | 0 | 0 |
| 60 | 30 | 0 | 0 |
| 43 | 30 | 0 | 0 |
| 0 | 0 | 30 | 43 |
| 0 | 0 | 30 | 60 |
| 0 | 0 | 43 | 30 |
| 0 | 0 | 43 | 13 |
| 0 | 27 | 0 | 0 |
| 46 | 46 | 0 | 0 |
G:=sub<GL(4,GF(73))| [7,7,0,0,66,14,0,0,0,0,7,7,0,0,66,14],[0,0,46,0,0,0,46,27,27,0,0,0,27,46,0,0],[60,43,0,0,30,30,0,0,0,0,30,30,0,0,43,60],[0,0,0,46,0,0,27,46,43,43,0,0,30,13,0,0] >;
D12.28D6 in GAP, Magma, Sage, TeX
D_{12}._{28}D_6 % in TeX
G:=Group("D12.28D6"); // GroupNames label
G:=SmallGroup(288,478);
// by ID
G=gap.SmallGroup(288,478);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,141,64,219,675,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^12=b^2=c^6=1,d^2=a^3,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=a^3*b,d*c*d^-1=a^6*c^-1>;
// generators/relations