metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.2Q16, C2.D8⋊4S3, D6⋊C8.7C2, C4⋊C4.49D6, (C2×C8).28D6, C6.24(C2×Q16), C2.14(S3×Q16), C6.Q16⋊22C2, C4.D12.7C2, C4.81(C4○D12), C12.39(C4○D4), C6.SD16⋊19C2, C2.22(D8⋊S3), C6.41(C8⋊C22), C2.Dic12⋊14C2, (C2×Dic3).49D4, (C22×S3).86D4, C22.230(S3×D4), (C2×C24).170C22, (C2×C12).300C23, C4.29(Q8⋊3S3), C3⋊3(C23.48D4), C2.16(D6.D4), C4⋊Dic3.125C22, (C2×Dic6).89C22, C6.46(C22.D4), (S3×C4⋊C4).9C2, (C3×C2.D8)⋊12C2, (C2×C6).305(C2×D4), (C2×C3⋊C8).70C22, (S3×C2×C4).39C22, (C3×C4⋊C4).93C22, (C2×C4).403(C22×S3), SmallGroup(192,443)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.2Q16
G = < a,b,c,d | a6=b2=c8=1, d2=a3c4, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=c-1 >
Subgroups: 304 in 104 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22⋊C8, Q8⋊C4, C2.D8, C2.D8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, S3×C2×C4, C23.48D4, C6.Q16, C6.SD16, C2.Dic12, D6⋊C8, C3×C2.D8, S3×C4⋊C4, C4.D12, D6.2Q16
Quotients: C1, C2, C22, S3, D4, C23, D6, Q16, C2×D4, C4○D4, C22×S3, C22.D4, C2×Q16, C8⋊C22, C4○D12, S3×D4, Q8⋊3S3, C23.48D4, D6.D4, D8⋊S3, S3×Q16, D6.2Q16
►On 96 points
(1 42 19 73 49 62)(2 43 20 74 50 63)(3 44 21 75 51 64)(4 45 22 76 52 57)(5 46 23 77 53 58)(6 47 24 78 54 59)(7 48 17 79 55 60)(8 41 18 80 56 61)(9 35 32 65 87 89)(10 36 25 66 88 90)(11 37 26 67 81 91)(12 38 27 68 82 92)(13 39 28 69 83 93)(14 40 29 70 84 94)(15 33 30 71 85 95)(16 34 31 72 86 96)
(1 58)(2 24)(3 60)(4 18)(5 62)(6 20)(7 64)(8 22)(9 93)(10 29)(11 95)(12 31)(13 89)(14 25)(15 91)(16 27)(17 75)(19 77)(21 79)(23 73)(26 71)(28 65)(30 67)(32 69)(33 81)(34 38)(35 83)(36 40)(37 85)(39 87)(41 45)(42 53)(43 47)(44 55)(46 49)(48 51)(50 54)(52 56)(57 80)(59 74)(61 76)(63 78)(66 94)(68 96)(70 90)(72 92)(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 13 77 65)(2 12 78 72)(3 11 79 71)(4 10 80 70)(5 9 73 69)(6 16 74 68)(7 15 75 67)(8 14 76 66)(17 30 64 91)(18 29 57 90)(19 28 58 89)(20 27 59 96)(21 26 60 95)(22 25 61 94)(23 32 62 93)(24 31 63 92)(33 51 81 48)(34 50 82 47)(35 49 83 46)(36 56 84 45)(37 55 85 44)(38 54 86 43)(39 53 87 42)(40 52 88 41)
G:=sub<Sym(96)| (1,42,19,73,49,62)(2,43,20,74,50,63)(3,44,21,75,51,64)(4,45,22,76,52,57)(5,46,23,77,53,58)(6,47,24,78,54,59)(7,48,17,79,55,60)(8,41,18,80,56,61)(9,35,32,65,87,89)(10,36,25,66,88,90)(11,37,26,67,81,91)(12,38,27,68,82,92)(13,39,28,69,83,93)(14,40,29,70,84,94)(15,33,30,71,85,95)(16,34,31,72,86,96), (1,58)(2,24)(3,60)(4,18)(5,62)(6,20)(7,64)(8,22)(9,93)(10,29)(11,95)(12,31)(13,89)(14,25)(15,91)(16,27)(17,75)(19,77)(21,79)(23,73)(26,71)(28,65)(30,67)(32,69)(33,81)(34,38)(35,83)(36,40)(37,85)(39,87)(41,45)(42,53)(43,47)(44,55)(46,49)(48,51)(50,54)(52,56)(57,80)(59,74)(61,76)(63,78)(66,94)(68,96)(70,90)(72,92)(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), (1,13,77,65)(2,12,78,72)(3,11,79,71)(4,10,80,70)(5,9,73,69)(6,16,74,68)(7,15,75,67)(8,14,76,66)(17,30,64,91)(18,29,57,90)(19,28,58,89)(20,27,59,96)(21,26,60,95)(22,25,61,94)(23,32,62,93)(24,31,63,92)(33,51,81,48)(34,50,82,47)(35,49,83,46)(36,56,84,45)(37,55,85,44)(38,54,86,43)(39,53,87,42)(40,52,88,41)>;
G:=Group( (1,42,19,73,49,62)(2,43,20,74,50,63)(3,44,21,75,51,64)(4,45,22,76,52,57)(5,46,23,77,53,58)(6,47,24,78,54,59)(7,48,17,79,55,60)(8,41,18,80,56,61)(9,35,32,65,87,89)(10,36,25,66,88,90)(11,37,26,67,81,91)(12,38,27,68,82,92)(13,39,28,69,83,93)(14,40,29,70,84,94)(15,33,30,71,85,95)(16,34,31,72,86,96), (1,58)(2,24)(3,60)(4,18)(5,62)(6,20)(7,64)(8,22)(9,93)(10,29)(11,95)(12,31)(13,89)(14,25)(15,91)(16,27)(17,75)(19,77)(21,79)(23,73)(26,71)(28,65)(30,67)(32,69)(33,81)(34,38)(35,83)(36,40)(37,85)(39,87)(41,45)(42,53)(43,47)(44,55)(46,49)(48,51)(50,54)(52,56)(57,80)(59,74)(61,76)(63,78)(66,94)(68,96)(70,90)(72,92)(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), (1,13,77,65)(2,12,78,72)(3,11,79,71)(4,10,80,70)(5,9,73,69)(6,16,74,68)(7,15,75,67)(8,14,76,66)(17,30,64,91)(18,29,57,90)(19,28,58,89)(20,27,59,96)(21,26,60,95)(22,25,61,94)(23,32,62,93)(24,31,63,92)(33,51,81,48)(34,50,82,47)(35,49,83,46)(36,56,84,45)(37,55,85,44)(38,54,86,43)(39,53,87,42)(40,52,88,41) );
G=PermutationGroup([[(1,42,19,73,49,62),(2,43,20,74,50,63),(3,44,21,75,51,64),(4,45,22,76,52,57),(5,46,23,77,53,58),(6,47,24,78,54,59),(7,48,17,79,55,60),(8,41,18,80,56,61),(9,35,32,65,87,89),(10,36,25,66,88,90),(11,37,26,67,81,91),(12,38,27,68,82,92),(13,39,28,69,83,93),(14,40,29,70,84,94),(15,33,30,71,85,95),(16,34,31,72,86,96)], [(1,58),(2,24),(3,60),(4,18),(5,62),(6,20),(7,64),(8,22),(9,93),(10,29),(11,95),(12,31),(13,89),(14,25),(15,91),(16,27),(17,75),(19,77),(21,79),(23,73),(26,71),(28,65),(30,67),(32,69),(33,81),(34,38),(35,83),(36,40),(37,85),(39,87),(41,45),(42,53),(43,47),(44,55),(46,49),(48,51),(50,54),(52,56),(57,80),(59,74),(61,76),(63,78),(66,94),(68,96),(70,90),(72,92),(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)], [(1,13,77,65),(2,12,78,72),(3,11,79,71),(4,10,80,70),(5,9,73,69),(6,16,74,68),(7,15,75,67),(8,14,76,66),(17,30,64,91),(18,29,57,90),(19,28,58,89),(20,27,59,96),(21,26,60,95),(22,25,61,94),(23,32,62,93),(24,31,63,92),(33,51,81,48),(34,50,82,47),(35,49,83,46),(36,56,84,45),(37,55,85,44),(38,54,86,43),(39,53,87,42),(40,52,88,41)]])
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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | C4○D4 | Q16 | C4○D12 | C8⋊C22 | Q8⋊3S3 | S3×D4 | D8⋊S3 | S3×Q16 |
| kernel | D6.2Q16 | C6.Q16 | C6.SD16 | C2.Dic12 | D6⋊C8 | C3×C2.D8 | S3×C4⋊C4 | C4.D12 | C2.D8 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C12 | D6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6.2Q16 ►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 |
| 30 | 60 | 0 | 0 |
| 13 | 43 | 0 | 0 |
| 0 | 0 | 51 | 48 |
| 0 | 0 | 0 | 63 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 7 | 34 |
| 0 | 0 | 20 | 66 |
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],[30,13,0,0,60,43,0,0,0,0,51,0,0,0,48,63],[46,0,0,0,0,46,0,0,0,0,7,20,0,0,34,66] >;
D6.2Q16 in GAP, Magma, Sage, TeX
D_6._2Q_{16} % in TeX
G:=Group("D6.2Q16"); // GroupNames label
G:=SmallGroup(192,443);
// by ID
G=gap.SmallGroup(192,443);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,926,219,268,851,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3*c^4,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations