metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.4D12, C24.83D4, Q8.9D12, M4(2).34D6, C8○D4⋊6S3, (C2×C8).81D6, D4⋊D6⋊3C2, (C2×D24)⋊13C2, C4○D4.48D6, (C3×D4).21D4, C4.20(C2×D12), C12.43(C2×D4), (C3×Q8).21D4, C3⋊3(D4.4D4), C8.40(C3⋊D4), C24.C4⋊15C2, C6.77(C4⋊D4), (C2×C24).67C22, C12.46D4⋊14C2, C2.25(C12⋊7D4), (C2×C12).422C23, C22.9(C4○D12), (C2×D12).112C22, C4.Dic3.17C22, (C3×M4(2)).37C22, (C3×C8○D4)⋊2C2, (C2×C6).7(C4○D4), C4.118(C2×C3⋊D4), (C2×C4).124(C22×S3), (C3×C4○D4).37C22, SmallGroup(192,701)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.9D12
G = < a,b,c,d | a4=d2=1, b2=c12=a2, bab-1=dad=a-1, ac=ca, bc=cb, dbd=ab, dcd=a2c11 >
Subgroups: 384 in 108 conjugacy classes, 39 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, D4, D4, Q8, C23, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, C2×D4, C4○D4, C3⋊C8, C24, C24, D12, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C8.C4, C8○D4, C2×D8, C8⋊C22, D24, C4.Dic3, D4⋊S3, Q8⋊2S3, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×D12, C3×C4○D4, D4.4D4, C24.C4, C12.46D4, C2×D24, D4⋊D6, C3×C8○D4, Q8.9D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C2×D12, C4○D12, C2×C3⋊D4, D4.4D4, C12⋊7D4, Q8.9D12
►On 48 points
(1 19 13 7)(2 20 14 8)(3 21 15 9)(4 22 16 10)(5 23 17 11)(6 24 18 12)(25 31 37 43)(26 32 38 44)(27 33 39 45)(28 34 40 46)(29 35 41 47)(30 36 42 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)
(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 48)(26 47)(27 46)(28 45)(29 44)(30 43)(31 42)(32 41)(33 40)(34 39)(35 38)(36 37)
G:=sub<Sym(48)| (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,31,37,43)(26,32,38,44)(27,33,39,45)(28,34,40,46)(29,35,41,47)(30,36,42,48), (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), (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)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)>;
G:=Group( (1,19,13,7)(2,20,14,8)(3,21,15,9)(4,22,16,10)(5,23,17,11)(6,24,18,12)(25,31,37,43)(26,32,38,44)(27,33,39,45)(28,34,40,46)(29,35,41,47)(30,36,42,48), (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), (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)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37) );
G=PermutationGroup([[(1,19,13,7),(2,20,14,8),(3,21,15,9),(4,22,16,10),(5,23,17,11),(6,24,18,12),(25,31,37,43),(26,32,38,44),(27,33,39,45),(28,34,40,46),(29,35,41,47),(30,36,42,48)], [(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)], [(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),(25,48),(26,47),(27,46),(28,45),(29,44),(30,43),(31,42),(32,41),(33,40),(34,39),(35,38),(36,37)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 12A | 12B | 12C | 12D | 12E | 24A | 24B | 24C | 24D | 24E | ··· | 24J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 2 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | D12 | C4○D12 | D4.4D4 | Q8.9D12 |
| kernel | Q8.9D12 | C24.C4 | C12.46D4 | C2×D24 | D4⋊D6 | C3×C8○D4 | C8○D4 | C24 | C3×D4 | C3×Q8 | C2×C8 | M4(2) | C4○D4 | C2×C6 | C8 | D4 | Q8 | C22 | C3 | C1 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of Q8.9D12 ►in GL4(𝔽73) generated by
| 7 | 14 | 0 | 0 |
| 59 | 66 | 0 | 0 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 0 | 0 | 66 | 59 |
| 0 | 0 | 14 | 7 |
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 23 | 18 | 0 | 0 |
| 55 | 5 | 0 | 0 |
| 0 | 0 | 23 | 18 |
| 0 | 0 | 55 | 5 |
| 23 | 18 | 0 | 0 |
| 68 | 50 | 0 | 0 |
| 0 | 0 | 55 | 50 |
| 0 | 0 | 68 | 18 |
G:=sub<GL(4,GF(73))| [7,59,0,0,14,66,0,0,0,0,66,14,0,0,59,7],[0,0,66,14,0,0,59,7,66,14,0,0,59,7,0,0],[23,55,0,0,18,5,0,0,0,0,23,55,0,0,18,5],[23,68,0,0,18,50,0,0,0,0,55,68,0,0,50,18] >;
Q8.9D12 in GAP, Magma, Sage, TeX
Q_8._9D_{12} % in TeX
G:=Group("Q8.9D12"); // GroupNames label
G:=SmallGroup(192,701);
// by ID
G=gap.SmallGroup(192,701);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,1123,297,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=d^2=1,b^2=c^12=a^2,b*a*b^-1=d*a*d=a^-1,a*c=c*a,b*c=c*b,d*b*d=a*b,d*c*d=a^2*c^11>;
// generators/relations