metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D12, Q8.17D12, D24⋊12C22, M4(2)⋊21D6, C12.62C24, C24.11C23, D12.25C23, Dic12⋊21C22, Dic6.25C23, (C2×C8)⋊7D6, C8○D4⋊8S3, C3⋊1(D4○D8), D4○D12⋊4C2, C4○D24⋊12C2, (C2×D24)⋊15C2, C8⋊D6⋊12C2, C4○D4.55D6, (C3×D4).24D4, C12.74(C2×D4), C4.28(C2×D12), (C3×Q8).24D4, (C2×C24)⋊10C22, C4○D12⋊2C22, C8.53(C22×S3), C4.59(S3×C23), C6.29(C22×D4), C22.4(C2×D12), C24⋊C2⋊12C22, (C2×D12)⋊31C22, C2.31(C22×D12), (C2×C12).516C23, (C3×M4(2))⋊23C22, (C3×C8○D4)⋊4C2, (C2×C6).9(C2×D4), (C2×C4).227(C22×S3), (C3×C4○D4).46C22, SmallGroup(192,1311)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D4.12D12
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c11 >
Subgroups: 880 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C2×C8, M4(2), D8, SD16, Q16, C2×D4, C4○D4, C4○D4, C24, C24, Dic6, C4×S3, D12, D12, C3⋊D4, C2×C12, C3×D4, C3×Q8, C22×S3, C8○D4, C2×D8, C4○D8, C8⋊C22, 2+ 1+4, C24⋊C2, D24, Dic12, C2×C24, C3×M4(2), C2×D12, C4○D12, S3×D4, Q8⋊3S3, C3×C4○D4, D4○D8, C2×D24, C4○D24, C8⋊D6, C3×C8○D4, D4○D12, D4.12D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C24, D12, C22×S3, C22×D4, C2×D12, S3×C23, D4○D8, C22×D12, D4.12D12
►On 48 points
(1 43 13 31)(2 44 14 32)(3 45 15 33)(4 46 16 34)(5 47 17 35)(6 48 18 36)(7 25 19 37)(8 26 20 38)(9 27 21 39)(10 28 22 40)(11 29 23 41)(12 30 24 42)
(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 24)(2 23)(3 22)(4 21)(5 20)(6 19)(7 18)(8 17)(9 16)(10 15)(11 14)(12 13)(25 36)(26 35)(27 34)(28 33)(29 32)(30 31)(37 48)(38 47)(39 46)(40 45)(41 44)(42 43)
G:=sub<Sym(48)| (1,43,13,31)(2,44,14,32)(3,45,15,33)(4,46,16,34)(5,47,17,35)(6,48,18,36)(7,25,19,37)(8,26,20,38)(9,27,21,39)(10,28,22,40)(11,29,23,41)(12,30,24,42), (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43)>;
G:=Group( (1,43,13,31)(2,44,14,32)(3,45,15,33)(4,46,16,34)(5,47,17,35)(6,48,18,36)(7,25,19,37)(8,26,20,38)(9,27,21,39)(10,28,22,40)(11,29,23,41)(12,30,24,42), (25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,24)(2,23)(3,22)(4,21)(5,20)(6,19)(7,18)(8,17)(9,16)(10,15)(11,14)(12,13)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(37,48)(38,47)(39,46)(40,45)(41,44)(42,43) );
G=PermutationGroup([[(1,43,13,31),(2,44,14,32),(3,45,15,33),(4,46,16,34),(5,47,17,35),(6,48,18,36),(7,25,19,37),(8,26,20,38),(9,27,21,39),(10,28,22,40),(11,29,23,41),(12,30,24,42)], [(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,24),(2,23),(3,22),(4,21),(5,20),(6,19),(7,18),(8,17),(9,16),(10,15),(11,14),(12,13),(25,36),(26,35),(27,34),(28,33),(29,32),(30,31),(37,48),(38,47),(39,46),(40,45),(41,44),(42,43)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | ··· | 2J | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 2 | 2 | 12 | ··· | 12 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + |
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | D12 | D12 | D4○D8 | D4.12D12 |
| kernel | D4.12D12 | C2×D24 | C4○D24 | C8⋊D6 | C3×C8○D4 | D4○D12 | C8○D4 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | D4 | Q8 | C3 | C1 |
| # reps | 1 | 3 | 3 | 6 | 1 | 2 | 1 | 3 | 1 | 3 | 3 | 1 | 6 | 2 | 2 | 4 |
Matrix representation of D4.12D12 ►in GL4(𝔽73) generated by
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 68 | 55 | 0 | 0 |
| 18 | 50 | 0 | 0 |
| 0 | 0 | 68 | 55 |
| 0 | 0 | 18 | 50 |
| 50 | 18 | 0 | 0 |
| 68 | 23 | 0 | 0 |
| 0 | 0 | 50 | 18 |
| 0 | 0 | 68 | 23 |
G:=sub<GL(4,GF(73))| [0,0,1,0,0,0,0,1,72,0,0,0,0,72,0,0],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,0,72],[68,18,0,0,55,50,0,0,0,0,68,18,0,0,55,50],[50,68,0,0,18,23,0,0,0,0,50,68,0,0,18,23] >;
D4.12D12 in GAP, Magma, Sage, TeX
D_4._{12}D_{12} % in TeX
G:=Group("D4.12D12"); // GroupNames label
G:=SmallGroup(192,1311);
// by ID
G=gap.SmallGroup(192,1311);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,387,675,192,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^2=d^2=1,c^12=a^2,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=a^2*c^11>;
// generators/relations