metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4.3D12, C24.87D4, Q8.8D12, M4(2).33D6, C8○D4⋊5S3, (C2×C8).80D6, C4○D4.47D6, (C3×D4).20D4, C12.42(C2×D4), C4.19(C2×D12), (C3×Q8).20D4, Q8.14D6⋊3C2, D4⋊D6.1C2, C3⋊4(D4.3D4), C8.44(C3⋊D4), C24.C4⋊14C2, C6.76(C4⋊D4), (C2×C24).66C22, C12.47D4⋊13C2, C12.46D4⋊13C2, C2.24(C12⋊7D4), (C2×C12).421C23, C22.8(C4○D12), (C2×D12).111C22, C4.Dic3.16C22, (C2×Dic6).117C22, (C3×M4(2)).36C22, (C3×C8○D4)⋊1C2, (C2×C24⋊C2)⋊3C2, (C2×C6).6(C4○D4), C4.117(C2×C3⋊D4), (C2×C4).123(C22×S3), (C3×C4○D4).36C22, SmallGroup(192,700)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8.8D12
G = < a,b,c,d | a4=1, b2=c12=d2=a2, bab-1=dad-1=a-1, ac=ca, bc=cb, dbd-1=ab, dcd-1=c11 >
Subgroups: 320 in 104 conjugacy classes, 39 normal (all characteristic)
C1, C2, C2, C3, 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×C6, C2×C8, C2×C8, M4(2), M4(2), D8, SD16, Q16, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C24, Dic6, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×S3, C4.D4, C4.10D4, C8.C4, C8○D4, C2×SD16, C8⋊C22, C8.C22, C24⋊C2, C4.Dic3, D4⋊S3, D4.S3, Q8⋊2S3, C3⋊Q16, C2×C24, C2×C24, C3×M4(2), C3×M4(2), C2×Dic6, C2×D12, C3×C4○D4, D4.3D4, C24.C4, C12.46D4, C12.47D4, C2×C24⋊C2, D4⋊D6, Q8.14D6, C3×C8○D4, Q8.8D12
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, D4.3D4, C12⋊7D4, Q8.8D12
►On 48 points
(1 7 13 19)(2 8 14 20)(3 9 15 21)(4 10 16 22)(5 11 17 23)(6 12 18 24)(25 43 37 31)(26 44 38 32)(27 45 39 33)(28 46 40 34)(29 47 41 35)(30 48 42 36)
(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)
(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 12 13 24)(2 23 14 11)(3 10 15 22)(4 21 16 9)(5 8 17 20)(6 19 18 7)(25 30 37 42)(26 41 38 29)(27 28 39 40)(31 48 43 36)(32 35 44 47)(33 46 45 34)
G:=sub<Sym(48)| (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36), (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), (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), (1,12,13,24)(2,23,14,11)(3,10,15,22)(4,21,16,9)(5,8,17,20)(6,19,18,7)(25,30,37,42)(26,41,38,29)(27,28,39,40)(31,48,43,36)(32,35,44,47)(33,46,45,34)>;
G:=Group( (1,7,13,19)(2,8,14,20)(3,9,15,21)(4,10,16,22)(5,11,17,23)(6,12,18,24)(25,43,37,31)(26,44,38,32)(27,45,39,33)(28,46,40,34)(29,47,41,35)(30,48,42,36), (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), (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), (1,12,13,24)(2,23,14,11)(3,10,15,22)(4,21,16,9)(5,8,17,20)(6,19,18,7)(25,30,37,42)(26,41,38,29)(27,28,39,40)(31,48,43,36)(32,35,44,47)(33,46,45,34) );
G=PermutationGroup([[(1,7,13,19),(2,8,14,20),(3,9,15,21),(4,10,16,22),(5,11,17,23),(6,12,18,24),(25,43,37,31),(26,44,38,32),(27,45,39,33),(28,46,40,34),(29,47,41,35),(30,48,42,36)], [(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)], [(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)], [(1,12,13,24),(2,23,14,11),(3,10,15,22),(4,21,16,9),(5,8,17,20),(6,19,18,7),(25,30,37,42),(26,41,38,29),(27,28,39,40),(31,48,43,36),(32,35,44,47),(33,46,45,34)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 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 | 3 | 4 | 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 | 2 | 2 | 2 | 4 | 24 | 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 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | D12 | C4○D12 | D4.3D4 | Q8.8D12 |
| kernel | Q8.8D12 | C24.C4 | C12.46D4 | C12.47D4 | C2×C24⋊C2 | D4⋊D6 | Q8.14D6 | 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 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of Q8.8D12 ►in GL4(𝔽73) generated by
| 66 | 59 | 0 | 0 |
| 14 | 7 | 0 | 0 |
| 46 | 0 | 7 | 14 |
| 0 | 46 | 59 | 66 |
| 27 | 0 | 59 | 45 |
| 0 | 27 | 28 | 14 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 48 | 37 | 0 | 0 |
| 36 | 11 | 0 | 0 |
| 0 | 0 | 48 | 37 |
| 0 | 0 | 36 | 11 |
| 48 | 37 | 0 | 0 |
| 62 | 25 | 0 | 0 |
| 16 | 16 | 36 | 25 |
| 0 | 57 | 62 | 37 |
G:=sub<GL(4,GF(73))| [66,14,46,0,59,7,0,46,0,0,7,59,0,0,14,66],[27,0,0,0,0,27,0,0,59,28,46,0,45,14,0,46],[48,36,0,0,37,11,0,0,0,0,48,36,0,0,37,11],[48,62,16,0,37,25,16,57,0,0,36,62,0,0,25,37] >;
Q8.8D12 in GAP, Magma, Sage, TeX
Q_8._8D_{12} % in TeX
G:=Group("Q8.8D12"); // GroupNames label
G:=SmallGroup(192,700);
// by ID
G=gap.SmallGroup(192,700);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,254,1123,297,136,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=1,b^2=c^12=d^2=a^2,b*a*b^-1=d*a*d^-1=a^-1,a*c=c*a,b*c=c*b,d*b*d^-1=a*b,d*c*d^-1=c^11>;
// generators/relations