metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊4Q16, Q8.12D12, Dic6.12D4, D6⋊C8.2C2, C4⋊C4.26D6, (C2×C8).16D6, C4.7(C2×D12), C4.94(S3×D4), (C3×Q8).2D4, C2.9(S3×Q16), Q8⋊C4⋊4S3, C6.17(C2×Q16), C6.25C22≀C2, (C2×Dic12)⋊6C2, C12.123(C2×D4), C4.D12.1C2, (C2×Q8).133D6, C3⋊2(C22⋊Q16), C6.SD16⋊12C2, (C2×C24).16C22, (C2×Dic3).31D4, (C22×S3).78D4, C22.199(S3×D4), (C6×Q8).32C22, C2.28(D6⋊D4), (C2×C12).249C23, C2.17(D4.D6), C6.35(C8.C22), (C2×Dic6).72C22, (C2×S3×Q8).4C2, (C2×C3⋊Q16)⋊4C2, (C3×Q8⋊C4)⋊4C2, (C2×C6).262(C2×D4), (C2×C3⋊C8).40C22, (S3×C2×C4).22C22, (C3×C4⋊C4).50C22, (C2×C4).356(C22×S3), SmallGroup(192,368)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊Q16
G = < a,b,c,d | a6=b2=c8=1, d2=c4, bab=cac-1=dad-1=a-1, cbc-1=dbd-1=ab, dcd-1=c-1 >
Subgroups: 424 in 148 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, Q16, C22×C4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C3×Q8, C22×S3, C22⋊C8, Q8⋊C4, Q8⋊C4, C22⋊Q8, C2×Q16, C22×Q8, Dic12, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3⋊Q16, C3×C4⋊C4, C2×C24, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C6×Q8, C22⋊Q16, C6.SD16, D6⋊C8, C3×Q8⋊C4, C4.D12, C2×Dic12, C2×C3⋊Q16, C2×S3×Q8, D6⋊Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, D12, C22×S3, C22≀C2, C2×Q16, C8.C22, C2×D12, S3×D4, C22⋊Q16, D6⋊D4, D4.D6, S3×Q16, D6⋊Q16
►On 96 points
(1 52 87 41 93 13)(2 14 94 42 88 53)(3 54 81 43 95 15)(4 16 96 44 82 55)(5 56 83 45 89 9)(6 10 90 46 84 49)(7 50 85 47 91 11)(8 12 92 48 86 51)(17 59 28 37 75 71)(18 72 76 38 29 60)(19 61 30 39 77 65)(20 66 78 40 31 62)(21 63 32 33 79 67)(22 68 80 34 25 64)(23 57 26 35 73 69)(24 70 74 36 27 58)
(1 9)(2 90)(3 11)(4 92)(5 13)(6 94)(7 15)(8 96)(10 14)(12 16)(17 32)(18 64)(19 26)(20 58)(21 28)(22 60)(23 30)(24 62)(25 72)(27 66)(29 68)(31 70)(33 71)(34 76)(35 65)(36 78)(37 67)(38 80)(39 69)(40 74)(41 83)(42 49)(43 85)(44 51)(45 87)(46 53)(47 81)(48 55)(50 95)(52 89)(54 91)(56 93)(57 61)(59 63)(73 77)(75 79)(82 86)(84 88)
(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 33 5 37)(2 40 6 36)(3 39 7 35)(4 38 8 34)(9 75 13 79)(10 74 14 78)(11 73 15 77)(12 80 16 76)(17 41 21 45)(18 48 22 44)(19 47 23 43)(20 46 24 42)(25 55 29 51)(26 54 30 50)(27 53 31 49)(28 52 32 56)(57 81 61 85)(58 88 62 84)(59 87 63 83)(60 86 64 82)(65 91 69 95)(66 90 70 94)(67 89 71 93)(68 96 72 92)
G:=sub<Sym(96)| (1,52,87,41,93,13)(2,14,94,42,88,53)(3,54,81,43,95,15)(4,16,96,44,82,55)(5,56,83,45,89,9)(6,10,90,46,84,49)(7,50,85,47,91,11)(8,12,92,48,86,51)(17,59,28,37,75,71)(18,72,76,38,29,60)(19,61,30,39,77,65)(20,66,78,40,31,62)(21,63,32,33,79,67)(22,68,80,34,25,64)(23,57,26,35,73,69)(24,70,74,36,27,58), (1,9)(2,90)(3,11)(4,92)(5,13)(6,94)(7,15)(8,96)(10,14)(12,16)(17,32)(18,64)(19,26)(20,58)(21,28)(22,60)(23,30)(24,62)(25,72)(27,66)(29,68)(31,70)(33,71)(34,76)(35,65)(36,78)(37,67)(38,80)(39,69)(40,74)(41,83)(42,49)(43,85)(44,51)(45,87)(46,53)(47,81)(48,55)(50,95)(52,89)(54,91)(56,93)(57,61)(59,63)(73,77)(75,79)(82,86)(84,88), (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), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,75,13,79)(10,74,14,78)(11,73,15,77)(12,80,16,76)(17,41,21,45)(18,48,22,44)(19,47,23,43)(20,46,24,42)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,52,32,56)(57,81,61,85)(58,88,62,84)(59,87,63,83)(60,86,64,82)(65,91,69,95)(66,90,70,94)(67,89,71,93)(68,96,72,92)>;
G:=Group( (1,52,87,41,93,13)(2,14,94,42,88,53)(3,54,81,43,95,15)(4,16,96,44,82,55)(5,56,83,45,89,9)(6,10,90,46,84,49)(7,50,85,47,91,11)(8,12,92,48,86,51)(17,59,28,37,75,71)(18,72,76,38,29,60)(19,61,30,39,77,65)(20,66,78,40,31,62)(21,63,32,33,79,67)(22,68,80,34,25,64)(23,57,26,35,73,69)(24,70,74,36,27,58), (1,9)(2,90)(3,11)(4,92)(5,13)(6,94)(7,15)(8,96)(10,14)(12,16)(17,32)(18,64)(19,26)(20,58)(21,28)(22,60)(23,30)(24,62)(25,72)(27,66)(29,68)(31,70)(33,71)(34,76)(35,65)(36,78)(37,67)(38,80)(39,69)(40,74)(41,83)(42,49)(43,85)(44,51)(45,87)(46,53)(47,81)(48,55)(50,95)(52,89)(54,91)(56,93)(57,61)(59,63)(73,77)(75,79)(82,86)(84,88), (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), (1,33,5,37)(2,40,6,36)(3,39,7,35)(4,38,8,34)(9,75,13,79)(10,74,14,78)(11,73,15,77)(12,80,16,76)(17,41,21,45)(18,48,22,44)(19,47,23,43)(20,46,24,42)(25,55,29,51)(26,54,30,50)(27,53,31,49)(28,52,32,56)(57,81,61,85)(58,88,62,84)(59,87,63,83)(60,86,64,82)(65,91,69,95)(66,90,70,94)(67,89,71,93)(68,96,72,92) );
G=PermutationGroup([[(1,52,87,41,93,13),(2,14,94,42,88,53),(3,54,81,43,95,15),(4,16,96,44,82,55),(5,56,83,45,89,9),(6,10,90,46,84,49),(7,50,85,47,91,11),(8,12,92,48,86,51),(17,59,28,37,75,71),(18,72,76,38,29,60),(19,61,30,39,77,65),(20,66,78,40,31,62),(21,63,32,33,79,67),(22,68,80,34,25,64),(23,57,26,35,73,69),(24,70,74,36,27,58)], [(1,9),(2,90),(3,11),(4,92),(5,13),(6,94),(7,15),(8,96),(10,14),(12,16),(17,32),(18,64),(19,26),(20,58),(21,28),(22,60),(23,30),(24,62),(25,72),(27,66),(29,68),(31,70),(33,71),(34,76),(35,65),(36,78),(37,67),(38,80),(39,69),(40,74),(41,83),(42,49),(43,85),(44,51),(45,87),(46,53),(47,81),(48,55),(50,95),(52,89),(54,91),(56,93),(57,61),(59,63),(73,77),(75,79),(82,86),(84,88)], [(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)], [(1,33,5,37),(2,40,6,36),(3,39,7,35),(4,38,8,34),(9,75,13,79),(10,74,14,78),(11,73,15,77),(12,80,16,76),(17,41,21,45),(18,48,22,44),(19,47,23,43),(20,46,24,42),(25,55,29,51),(26,54,30,50),(27,53,31,49),(28,52,32,56),(57,81,61,85),(58,88,62,84),(59,87,63,83),(60,86,64,82),(65,91,69,95),(66,90,70,94),(67,89,71,93),(68,96,72,92)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 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 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 12 | 24 | 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 | Q16 | D12 | C8.C22 | S3×D4 | S3×D4 | D4.D6 | S3×Q16 |
| kernel | D6⋊Q16 | C6.SD16 | D6⋊C8 | C3×Q8⋊C4 | C4.D12 | C2×Dic12 | C2×C3⋊Q16 | C2×S3×Q8 | Q8⋊C4 | Dic6 | C2×Dic3 | C3×Q8 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | D6 | Q8 | 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 D6⋊Q16 ►in GL4(𝔽73) generated by
| 1 | 1 | 0 | 0 |
| 72 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 1 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 7 | 14 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 0 | 41 |
| 0 | 0 | 16 | 41 |
| 7 | 14 | 0 | 0 |
| 7 | 66 | 0 | 0 |
| 0 | 0 | 40 | 31 |
| 0 | 0 | 19 | 33 |
G:=sub<GL(4,GF(73))| [1,72,0,0,1,0,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,1,72,0,0,0,0,72,0,0,0,0,72],[7,7,0,0,14,66,0,0,0,0,0,16,0,0,41,41],[7,7,0,0,14,66,0,0,0,0,40,19,0,0,31,33] >;
D6⋊Q16 in GAP, Magma, Sage, TeX
D_6\rtimes Q_{16} % in TeX
G:=Group("D6:Q16"); // GroupNames label
G:=SmallGroup(192,368);
// by ID
G=gap.SmallGroup(192,368);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,226,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=c^4,b*a*b=c*a*c^-1=d*a*d^-1=a^-1,c*b*c^-1=d*b*d^-1=a*b,d*c*d^-1=c^-1>;
// generators/relations