metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.3SD16, D6⋊C8⋊10C2, C4⋊C4.136D6, C8⋊Dic3⋊12C2, (C2×D4).25D6, (C2×C8).115D6, D4⋊C4⋊11S3, D6⋊3D4.2C2, D4⋊Dic3⋊8C2, C12.Q8⋊4C2, C6.24(C2×SD16), C2.12(S3×SD16), C4.52(C4○D12), C2.14(D8⋊S3), C6.32(C8⋊C22), (C2×Dic3).21D4, (C6×D4).38C22, (C22×S3).73D4, C22.175(S3×D4), C12.150(C4○D4), C4.79(D4⋊2S3), (C2×C24).126C22, (C2×C12).217C23, C3⋊2(C23.46D4), C4⋊Dic3.71C22, C2.13(C23.9D6), C6.21(C22.D4), (S3×C4⋊C4)⋊4C2, (C2×C6).230(C2×D4), (C2×C3⋊C8).16C22, (S3×C2×C4).10C22, (C3×D4⋊C4)⋊11C2, (C3×C4⋊C4).18C22, (C2×C4).324(C22×S3), SmallGroup(192,336)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=a3c3 >
Subgroups: 360 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, C22×C4, C2×D4, C2×D4, C3⋊C8, C24, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, C22⋊C8, D4⋊C4, D4⋊C4, C4.Q8, C2×C4⋊C4, C4⋊D4, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, C6.D4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C2×C3⋊D4, C6×D4, C23.46D4, C12.Q8, C8⋊Dic3, D6⋊C8, D4⋊Dic3, C3×D4⋊C4, S3×C4⋊C4, D6⋊3D4, D6.SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C22×S3, C22.D4, C2×SD16, C8⋊C22, C4○D12, S3×D4, D4⋊2S3, C23.46D4, C23.9D6, D8⋊S3, S3×SD16, D6.SD16
►On 96 points
(1 56 65 81 38 14)(2 49 66 82 39 15)(3 50 67 83 40 16)(4 51 68 84 33 9)(5 52 69 85 34 10)(6 53 70 86 35 11)(7 54 71 87 36 12)(8 55 72 88 37 13)(17 42 32 57 95 77)(18 43 25 58 96 78)(19 44 26 59 89 79)(20 45 27 60 90 80)(21 46 28 61 91 73)(22 47 29 62 92 74)(23 48 30 63 93 75)(24 41 31 64 94 76)
(1 76)(2 32)(3 78)(4 26)(5 80)(6 28)(7 74)(8 30)(9 59)(10 20)(11 61)(12 22)(13 63)(14 24)(15 57)(16 18)(17 66)(19 68)(21 70)(23 72)(25 83)(27 85)(29 87)(31 81)(33 89)(34 45)(35 91)(36 47)(37 93)(38 41)(39 95)(40 43)(42 49)(44 51)(46 53)(48 55)(50 96)(52 90)(54 92)(56 94)(58 67)(60 69)(62 71)(64 65)(73 86)(75 88)(77 82)(79 84)
(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)
(2 84)(3 7)(4 82)(6 88)(8 86)(9 66)(11 72)(12 16)(13 70)(15 68)(17 19)(18 62)(20 60)(21 23)(22 58)(24 64)(25 74)(26 32)(27 80)(28 30)(29 78)(31 76)(33 49)(35 55)(36 40)(37 53)(39 51)(41 94)(42 44)(43 92)(45 90)(46 48)(47 96)(50 54)(57 59)(61 63)(67 71)(73 75)(77 79)(83 87)(89 95)(91 93)
G:=sub<Sym(96)| (1,56,65,81,38,14)(2,49,66,82,39,15)(3,50,67,83,40,16)(4,51,68,84,33,9)(5,52,69,85,34,10)(6,53,70,86,35,11)(7,54,71,87,36,12)(8,55,72,88,37,13)(17,42,32,57,95,77)(18,43,25,58,96,78)(19,44,26,59,89,79)(20,45,27,60,90,80)(21,46,28,61,91,73)(22,47,29,62,92,74)(23,48,30,63,93,75)(24,41,31,64,94,76), (1,76)(2,32)(3,78)(4,26)(5,80)(6,28)(7,74)(8,30)(9,59)(10,20)(11,61)(12,22)(13,63)(14,24)(15,57)(16,18)(17,66)(19,68)(21,70)(23,72)(25,83)(27,85)(29,87)(31,81)(33,89)(34,45)(35,91)(36,47)(37,93)(38,41)(39,95)(40,43)(42,49)(44,51)(46,53)(48,55)(50,96)(52,90)(54,92)(56,94)(58,67)(60,69)(62,71)(64,65)(73,86)(75,88)(77,82)(79,84), (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), (2,84)(3,7)(4,82)(6,88)(8,86)(9,66)(11,72)(12,16)(13,70)(15,68)(17,19)(18,62)(20,60)(21,23)(22,58)(24,64)(25,74)(26,32)(27,80)(28,30)(29,78)(31,76)(33,49)(35,55)(36,40)(37,53)(39,51)(41,94)(42,44)(43,92)(45,90)(46,48)(47,96)(50,54)(57,59)(61,63)(67,71)(73,75)(77,79)(83,87)(89,95)(91,93)>;
G:=Group( (1,56,65,81,38,14)(2,49,66,82,39,15)(3,50,67,83,40,16)(4,51,68,84,33,9)(5,52,69,85,34,10)(6,53,70,86,35,11)(7,54,71,87,36,12)(8,55,72,88,37,13)(17,42,32,57,95,77)(18,43,25,58,96,78)(19,44,26,59,89,79)(20,45,27,60,90,80)(21,46,28,61,91,73)(22,47,29,62,92,74)(23,48,30,63,93,75)(24,41,31,64,94,76), (1,76)(2,32)(3,78)(4,26)(5,80)(6,28)(7,74)(8,30)(9,59)(10,20)(11,61)(12,22)(13,63)(14,24)(15,57)(16,18)(17,66)(19,68)(21,70)(23,72)(25,83)(27,85)(29,87)(31,81)(33,89)(34,45)(35,91)(36,47)(37,93)(38,41)(39,95)(40,43)(42,49)(44,51)(46,53)(48,55)(50,96)(52,90)(54,92)(56,94)(58,67)(60,69)(62,71)(64,65)(73,86)(75,88)(77,82)(79,84), (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), (2,84)(3,7)(4,82)(6,88)(8,86)(9,66)(11,72)(12,16)(13,70)(15,68)(17,19)(18,62)(20,60)(21,23)(22,58)(24,64)(25,74)(26,32)(27,80)(28,30)(29,78)(31,76)(33,49)(35,55)(36,40)(37,53)(39,51)(41,94)(42,44)(43,92)(45,90)(46,48)(47,96)(50,54)(57,59)(61,63)(67,71)(73,75)(77,79)(83,87)(89,95)(91,93) );
G=PermutationGroup([[(1,56,65,81,38,14),(2,49,66,82,39,15),(3,50,67,83,40,16),(4,51,68,84,33,9),(5,52,69,85,34,10),(6,53,70,86,35,11),(7,54,71,87,36,12),(8,55,72,88,37,13),(17,42,32,57,95,77),(18,43,25,58,96,78),(19,44,26,59,89,79),(20,45,27,60,90,80),(21,46,28,61,91,73),(22,47,29,62,92,74),(23,48,30,63,93,75),(24,41,31,64,94,76)], [(1,76),(2,32),(3,78),(4,26),(5,80),(6,28),(7,74),(8,30),(9,59),(10,20),(11,61),(12,22),(13,63),(14,24),(15,57),(16,18),(17,66),(19,68),(21,70),(23,72),(25,83),(27,85),(29,87),(31,81),(33,89),(34,45),(35,91),(36,47),(37,93),(38,41),(39,95),(40,43),(42,49),(44,51),(46,53),(48,55),(50,96),(52,90),(54,92),(56,94),(58,67),(60,69),(62,71),(64,65),(73,86),(75,88),(77,82),(79,84)], [(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)], [(2,84),(3,7),(4,82),(6,88),(8,86),(9,66),(11,72),(12,16),(13,70),(15,68),(17,19),(18,62),(20,60),(21,23),(22,58),(24,64),(25,74),(26,32),(27,80),(28,30),(29,78),(31,76),(33,49),(35,55),(36,40),(37,53),(39,51),(41,94),(42,44),(43,92),(45,90),(46,48),(47,96),(50,54),(57,59),(61,63),(67,71),(73,75),(77,79),(83,87),(89,95),(91,93)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 8 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 12 | 24 | 2 | 2 | 2 | 8 | 8 | 4 | 4 | 12 | 12 | 4 | 4 | 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 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D6 | D6 | D6 | C4○D4 | SD16 | C4○D12 | C8⋊C22 | D4⋊2S3 | S3×D4 | D8⋊S3 | S3×SD16 |
| kernel | D6.SD16 | C12.Q8 | C8⋊Dic3 | D6⋊C8 | D4⋊Dic3 | C3×D4⋊C4 | S3×C4⋊C4 | D6⋊3D4 | D4⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×D4 | C12 | D6 | C4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of D6.SD16 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 9 | 0 |
| 0 | 0 | 69 | 65 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 57 |
| 0 | 0 | 53 | 51 |
| 61 | 18 | 0 | 0 |
| 69 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 56 | 46 |
| 1 | 0 | 0 | 0 |
| 25 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 21 | 72 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,9,69,0,0,0,65],[1,0,0,0,0,1,0,0,0,0,22,53,0,0,57,51],[61,69,0,0,18,0,0,0,0,0,27,56,0,0,0,46],[1,25,0,0,0,72,0,0,0,0,1,21,0,0,0,72] >;
D6.SD16 in GAP, Magma, Sage, TeX
D_6.{\rm SD}_{16} % in TeX
G:=Group("D6.SD16"); // GroupNames label
G:=SmallGroup(192,336);
// by ID
G=gap.SmallGroup(192,336);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,926,219,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=d^2=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=a^3*c^3>;
// generators/relations