metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.12D12, Q8.17D12, D24⋊12C22, M4(2)⋊21D6, C24.11C23, C12.62C24, 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, C4.28(C2×D12), C12.74(C2×D4), (C3×Q8).24D4, (C2×C24)⋊10C22, C4○D12⋊2C22, C4.59(S3×C23), C8.53(C22×S3), 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
Subgroups: 880 in 268 conjugacy classes, 107 normal (16 characteristic)
C1, C2, C2 [×9], C3, C4, C4 [×3], C4 [×2], C22 [×3], C22 [×12], S3 [×6], C6, C6 [×3], C8, C8 [×3], C2×C4 [×3], C2×C4 [×6], D4 [×3], D4 [×18], Q8, Q8 [×2], C23 [×6], Dic3 [×2], C12, C12 [×3], D6 [×12], C2×C6 [×3], C2×C8 [×3], M4(2) [×3], D8 [×9], SD16 [×6], Q16, C2×D4 [×12], C4○D4, C4○D4 [×8], C24, C24 [×3], Dic6 [×2], C4×S3 [×6], D12 [×6], D12 [×6], C3⋊D4 [×6], C2×C12 [×3], C3×D4 [×3], C3×Q8, C22×S3 [×6], C8○D4, C2×D8 [×3], C4○D8 [×3], C8⋊C22 [×6], 2+ (1+4) [×2], C24⋊C2 [×6], D24 [×9], Dic12, C2×C24 [×3], C3×M4(2) [×3], C2×D12 [×6], C4○D12 [×6], S3×D4 [×6], Q8⋊3S3 [×2], C3×C4○D4, D4○D8, C2×D24 [×3], C4○D24 [×3], C8⋊D6 [×6], C3×C8○D4, D4○D12 [×2], D4.12D12
Quotients:
C1, C2 [×15], C22 [×35], S3, D4 [×4], C23 [×15], D6 [×7], C2×D4 [×6], C24, D12 [×4], C22×S3 [×7], C22×D4, C2×D12 [×6], S3×C23, D4○D8, C22×D12, D4.12D12
Generators and relations
G = < a,b,c,d | a4=b2=d2=1, c12=a2, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a2c11 >
►On 48 points
(1 40 13 28)(2 41 14 29)(3 42 15 30)(4 43 16 31)(5 44 17 32)(6 45 18 33)(7 46 19 34)(8 47 20 35)(9 48 21 36)(10 25 22 37)(11 26 23 38)(12 27 24 39)
(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 30)(26 29)(27 28)(31 48)(32 47)(33 46)(34 45)(35 44)(36 43)(37 42)(38 41)(39 40)
G:=sub<Sym(48)| (1,40,13,28)(2,41,14,29)(3,42,15,30)(4,43,16,31)(5,44,17,32)(6,45,18,33)(7,46,19,34)(8,47,20,35)(9,48,21,36)(10,25,22,37)(11,26,23,38)(12,27,24,39), (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)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40)>;
G:=Group( (1,40,13,28)(2,41,14,29)(3,42,15,30)(4,43,16,31)(5,44,17,32)(6,45,18,33)(7,46,19,34)(8,47,20,35)(9,48,21,36)(10,25,22,37)(11,26,23,38)(12,27,24,39), (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)(25,30)(26,29)(27,28)(31,48)(32,47)(33,46)(34,45)(35,44)(36,43)(37,42)(38,41)(39,40) );
G=PermutationGroup([(1,40,13,28),(2,41,14,29),(3,42,15,30),(4,43,16,31),(5,44,17,32),(6,45,18,33),(7,46,19,34),(8,47,20,35),(9,48,21,36),(10,25,22,37),(11,26,23,38),(12,27,24,39)], [(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),(25,30),(26,29),(27,28),(31,48),(32,47),(33,46),(34,45),(35,44),(36,43),(37,42),(38,41),(39,40)])
Matrix representation ►G ⊆ 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] >;
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 |
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