metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12⋊4D4, Q8⋊4D12, D6⋊C8⋊8C2, (C3×Q8)⋊2D4, (C2×D24)⋊6C2, C4⋊C4.27D6, C4.8(C2×D12), (C2×C8).17D6, C4.95(S3×D4), C12⋊D4⋊5C2, C3⋊3(D4⋊D4), Q8⋊C4⋊5S3, C6.26C22≀C2, C6.70(C4○D8), C12.124(C2×D4), C6.D8⋊12C2, (C2×Q8).134D6, C2.16(Q8⋊3D6), C6.62(C8⋊C22), (C2×C24).17C22, (C22×S3).18D4, C22.200(S3×D4), (C6×Q8).33C22, C2.29(D6⋊D4), C2.9(D24⋊C2), (C2×C12).250C23, (C2×Dic3).155D4, (C2×D12).65C22, (C2×Q8⋊2S3)⋊4C2, (C3×Q8⋊C4)⋊5C2, (C2×Q8⋊3S3)⋊1C2, (C2×C6).263(C2×D4), (C2×C3⋊C8).41C22, (S3×C2×C4).23C22, (C3×C4⋊C4).51C22, (C2×C4).357(C22×S3), SmallGroup(192,369)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Q8⋊4D12
G = < a,b,c,d | a4=c12=d2=1, b2=a2, bab-1=cac-1=dad=a-1, cbc-1=a-1b, dbd=ab, dcd=c-1 >
Subgroups: 584 in 162 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, D8, SD16, C22×C4, C2×D4, C2×Q8, C4○D4, C3⋊C8, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22×S3, C22⋊C8, D4⋊C4, Q8⋊C4, C4⋊D4, C2×D8, C2×SD16, C2×C4○D4, D24, C2×C3⋊C8, D6⋊C4, Q8⋊2S3, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×D12, C2×D12, Q8⋊3S3, C6×Q8, D4⋊D4, C6.D8, D6⋊C8, C3×Q8⋊C4, C12⋊D4, C2×D24, C2×Q8⋊2S3, C2×Q8⋊3S3, Q8⋊4D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, D12, C22×S3, C22≀C2, C4○D8, C8⋊C22, C2×D12, S3×D4, D4⋊D4, D6⋊D4, Q8⋊3D6, D24⋊C2, Q8⋊4D12
►On 96 points
(1 14 85 64)(2 65 86 15)(3 16 87 66)(4 67 88 17)(5 18 89 68)(6 69 90 19)(7 20 91 70)(8 71 92 21)(9 22 93 72)(10 61 94 23)(11 24 95 62)(12 63 96 13)(25 81 57 46)(26 47 58 82)(27 83 59 48)(28 37 60 84)(29 73 49 38)(30 39 50 74)(31 75 51 40)(32 41 52 76)(33 77 53 42)(34 43 54 78)(35 79 55 44)(36 45 56 80)
(1 45 85 80)(2 57 86 25)(3 47 87 82)(4 59 88 27)(5 37 89 84)(6 49 90 29)(7 39 91 74)(8 51 92 31)(9 41 93 76)(10 53 94 33)(11 43 95 78)(12 55 96 35)(13 44 63 79)(14 36 64 56)(15 46 65 81)(16 26 66 58)(17 48 67 83)(18 28 68 60)(19 38 69 73)(20 30 70 50)(21 40 71 75)(22 32 72 52)(23 42 61 77)(24 34 62 54)
(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)(49 50 51 52 53 54 55 56 57 58 59 60)(61 62 63 64 65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96)
(1 3)(4 12)(5 11)(6 10)(7 9)(13 67)(14 66)(15 65)(16 64)(17 63)(18 62)(19 61)(20 72)(21 71)(22 70)(23 69)(24 68)(25 46)(26 45)(27 44)(28 43)(29 42)(30 41)(31 40)(32 39)(33 38)(34 37)(35 48)(36 47)(49 77)(50 76)(51 75)(52 74)(53 73)(54 84)(55 83)(56 82)(57 81)(58 80)(59 79)(60 78)(85 87)(88 96)(89 95)(90 94)(91 93)
G:=sub<Sym(96)| (1,14,85,64)(2,65,86,15)(3,16,87,66)(4,67,88,17)(5,18,89,68)(6,69,90,19)(7,20,91,70)(8,71,92,21)(9,22,93,72)(10,61,94,23)(11,24,95,62)(12,63,96,13)(25,81,57,46)(26,47,58,82)(27,83,59,48)(28,37,60,84)(29,73,49,38)(30,39,50,74)(31,75,51,40)(32,41,52,76)(33,77,53,42)(34,43,54,78)(35,79,55,44)(36,45,56,80), (1,45,85,80)(2,57,86,25)(3,47,87,82)(4,59,88,27)(5,37,89,84)(6,49,90,29)(7,39,91,74)(8,51,92,31)(9,41,93,76)(10,53,94,33)(11,43,95,78)(12,55,96,35)(13,44,63,79)(14,36,64,56)(15,46,65,81)(16,26,66,58)(17,48,67,83)(18,28,68,60)(19,38,69,73)(20,30,70,50)(21,40,71,75)(22,32,72,52)(23,42,61,77)(24,34,62,54), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,3)(4,12)(5,11)(6,10)(7,9)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,72)(21,71)(22,70)(23,69)(24,68)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,77)(50,76)(51,75)(52,74)(53,73)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(85,87)(88,96)(89,95)(90,94)(91,93)>;
G:=Group( (1,14,85,64)(2,65,86,15)(3,16,87,66)(4,67,88,17)(5,18,89,68)(6,69,90,19)(7,20,91,70)(8,71,92,21)(9,22,93,72)(10,61,94,23)(11,24,95,62)(12,63,96,13)(25,81,57,46)(26,47,58,82)(27,83,59,48)(28,37,60,84)(29,73,49,38)(30,39,50,74)(31,75,51,40)(32,41,52,76)(33,77,53,42)(34,43,54,78)(35,79,55,44)(36,45,56,80), (1,45,85,80)(2,57,86,25)(3,47,87,82)(4,59,88,27)(5,37,89,84)(6,49,90,29)(7,39,91,74)(8,51,92,31)(9,41,93,76)(10,53,94,33)(11,43,95,78)(12,55,96,35)(13,44,63,79)(14,36,64,56)(15,46,65,81)(16,26,66,58)(17,48,67,83)(18,28,68,60)(19,38,69,73)(20,30,70,50)(21,40,71,75)(22,32,72,52)(23,42,61,77)(24,34,62,54), (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)(49,50,51,52,53,54,55,56,57,58,59,60)(61,62,63,64,65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96), (1,3)(4,12)(5,11)(6,10)(7,9)(13,67)(14,66)(15,65)(16,64)(17,63)(18,62)(19,61)(20,72)(21,71)(22,70)(23,69)(24,68)(25,46)(26,45)(27,44)(28,43)(29,42)(30,41)(31,40)(32,39)(33,38)(34,37)(35,48)(36,47)(49,77)(50,76)(51,75)(52,74)(53,73)(54,84)(55,83)(56,82)(57,81)(58,80)(59,79)(60,78)(85,87)(88,96)(89,95)(90,94)(91,93) );
G=PermutationGroup([[(1,14,85,64),(2,65,86,15),(3,16,87,66),(4,67,88,17),(5,18,89,68),(6,69,90,19),(7,20,91,70),(8,71,92,21),(9,22,93,72),(10,61,94,23),(11,24,95,62),(12,63,96,13),(25,81,57,46),(26,47,58,82),(27,83,59,48),(28,37,60,84),(29,73,49,38),(30,39,50,74),(31,75,51,40),(32,41,52,76),(33,77,53,42),(34,43,54,78),(35,79,55,44),(36,45,56,80)], [(1,45,85,80),(2,57,86,25),(3,47,87,82),(4,59,88,27),(5,37,89,84),(6,49,90,29),(7,39,91,74),(8,51,92,31),(9,41,93,76),(10,53,94,33),(11,43,95,78),(12,55,96,35),(13,44,63,79),(14,36,64,56),(15,46,65,81),(16,26,66,58),(17,48,67,83),(18,28,68,60),(19,38,69,73),(20,30,70,50),(21,40,71,75),(22,32,72,52),(23,42,61,77),(24,34,62,54)], [(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),(49,50,51,52,53,54,55,56,57,58,59,60),(61,62,63,64,65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96)], [(1,3),(4,12),(5,11),(6,10),(7,9),(13,67),(14,66),(15,65),(16,64),(17,63),(18,62),(19,61),(20,72),(21,71),(22,70),(23,69),(24,68),(25,46),(26,45),(27,44),(28,43),(29,42),(30,41),(31,40),(32,39),(33,38),(34,37),(35,48),(36,47),(49,77),(50,76),(51,75),(52,74),(53,73),(54,84),(55,83),(56,82),(57,81),(58,80),(59,79),(60,78),(85,87),(88,96),(89,95),(90,94),(91,93)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 8 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | D12 | C4○D8 | C8⋊C22 | S3×D4 | S3×D4 | Q8⋊3D6 | D24⋊C2 |
| kernel | Q8⋊4D12 | C6.D8 | D6⋊C8 | C3×Q8⋊C4 | C12⋊D4 | C2×D24 | C2×Q8⋊2S3 | C2×Q8⋊3S3 | Q8⋊C4 | D12 | C2×Dic3 | C3×Q8 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | Q8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 2 | 1 | 1 | 1 | 1 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Q8⋊4D12 ►in GL4(𝔽73) generated by
| 0 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 67 | 6 | 0 | 0 |
| 6 | 6 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 0 | 46 | 0 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 0 | 59 | 7 |
| 0 | 0 | 66 | 66 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [0,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[67,6,0,0,6,6,0,0,0,0,72,0,0,0,0,72],[0,46,0,0,46,0,0,0,0,0,59,66,0,0,7,66],[72,0,0,0,0,1,0,0,0,0,72,0,0,0,1,1] >;
Q8⋊4D12 in GAP, Magma, Sage, TeX
Q_8\rtimes_4D_{12} % in TeX
G:=Group("Q8:4D12"); // GroupNames label
G:=SmallGroup(192,369);
// by ID
G=gap.SmallGroup(192,369);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,758,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^4=c^12=d^2=1,b^2=a^2,b*a*b^-1=c*a*c^-1=d*a*d=a^-1,c*b*c^-1=a^-1*b,d*b*d=a*b,d*c*d=c^-1>;
// generators/relations