metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6⋊8SD16, Dic6⋊7D4, (C3×Q8)⋊5D4, D6⋊C8⋊33C2, C4.63(S3×D4), (C2×D4).72D6, Q8⋊3(C3⋊D4), C3⋊4(Q8⋊D4), (C2×C8).147D6, C12.48(C2×D4), C6.58C22≀C2, (C6×SD16)⋊21C2, (C2×SD16)⋊11S3, D6⋊3D4.7C2, (C2×Q8).140D6, C2.29(S3×SD16), C6.46(C2×SD16), Q8⋊2Dic3⋊28C2, C2.Dic12⋊35C2, (C2×Dic3).72D4, (C6×D4).96C22, (C22×S3).92D4, C22.267(S3×D4), (C6×Q8).76C22, C2.26(C23⋊2D6), (C2×C12).447C23, (C2×C24).294C22, C2.29(D4.D6), C6.49(C8.C22), C4⋊Dic3.174C22, (C2×Dic6).127C22, (C2×S3×Q8)⋊2C2, C4.43(C2×C3⋊D4), (C2×D4.S3)⋊20C2, (C2×C6).359(C2×D4), (S3×C2×C4).48C22, (C2×C3⋊C8).157C22, (C2×C4).536(C22×S3), SmallGroup(192,729)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6⋊8SD16
G = < a,b,c,d | a6=b2=c8=d2=1, bab=a-1, ac=ca, ad=da, cbc-1=dbd=a3b, dcd=c3 >
Subgroups: 472 in 158 conjugacy classes, 45 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, Q8, Q8, C23, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, C2×Dic3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×Q8, C3×Q8, C22×S3, C22×C6, C22⋊C8, Q8⋊C4, C4⋊D4, C2×SD16, C2×SD16, C22×Q8, C2×C3⋊C8, C4⋊Dic3, D4.S3, C6.D4, C2×C24, C3×SD16, C2×Dic6, C2×Dic6, S3×C2×C4, S3×C2×C4, S3×Q8, C2×C3⋊D4, C6×D4, C6×Q8, Q8⋊D4, C2.Dic12, D6⋊C8, Q8⋊2Dic3, C2×D4.S3, D6⋊3D4, C6×SD16, C2×S3×Q8, D6⋊8SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C3⋊D4, C22×S3, C22≀C2, C2×SD16, C8.C22, S3×D4, C2×C3⋊D4, Q8⋊D4, S3×SD16, D4.D6, C23⋊2D6, D6⋊8SD16
►On 96 points
(1 42 19 77 70 54)(2 43 20 78 71 55)(3 44 21 79 72 56)(4 45 22 80 65 49)(5 46 23 73 66 50)(6 47 24 74 67 51)(7 48 17 75 68 52)(8 41 18 76 69 53)(9 96 83 63 28 35)(10 89 84 64 29 36)(11 90 85 57 30 37)(12 91 86 58 31 38)(13 92 87 59 32 39)(14 93 88 60 25 40)(15 94 81 61 26 33)(16 95 82 62 27 34)
(1 37)(2 86)(3 39)(4 88)(5 33)(6 82)(7 35)(8 84)(9 52)(10 18)(11 54)(12 20)(13 56)(14 22)(15 50)(16 24)(17 63)(19 57)(21 59)(23 61)(25 65)(26 46)(27 67)(28 48)(29 69)(30 42)(31 71)(32 44)(34 74)(36 76)(38 78)(40 80)(41 89)(43 91)(45 93)(47 95)(49 60)(51 62)(53 64)(55 58)(66 94)(68 96)(70 90)(72 92)(73 81)(75 83)(77 85)(79 87)
(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 4)(3 7)(6 8)(9 59)(10 62)(11 57)(12 60)(13 63)(14 58)(15 61)(16 64)(17 21)(18 24)(20 22)(25 91)(26 94)(27 89)(28 92)(29 95)(30 90)(31 93)(32 96)(33 81)(34 84)(35 87)(36 82)(37 85)(38 88)(39 83)(40 86)(41 47)(43 45)(44 48)(49 55)(51 53)(52 56)(65 71)(67 69)(68 72)(74 76)(75 79)(78 80)
G:=sub<Sym(96)| (1,42,19,77,70,54)(2,43,20,78,71,55)(3,44,21,79,72,56)(4,45,22,80,65,49)(5,46,23,73,66,50)(6,47,24,74,67,51)(7,48,17,75,68,52)(8,41,18,76,69,53)(9,96,83,63,28,35)(10,89,84,64,29,36)(11,90,85,57,30,37)(12,91,86,58,31,38)(13,92,87,59,32,39)(14,93,88,60,25,40)(15,94,81,61,26,33)(16,95,82,62,27,34), (1,37)(2,86)(3,39)(4,88)(5,33)(6,82)(7,35)(8,84)(9,52)(10,18)(11,54)(12,20)(13,56)(14,22)(15,50)(16,24)(17,63)(19,57)(21,59)(23,61)(25,65)(26,46)(27,67)(28,48)(29,69)(30,42)(31,71)(32,44)(34,74)(36,76)(38,78)(40,80)(41,89)(43,91)(45,93)(47,95)(49,60)(51,62)(53,64)(55,58)(66,94)(68,96)(70,90)(72,92)(73,81)(75,83)(77,85)(79,87), (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), (2,4)(3,7)(6,8)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,21)(18,24)(20,22)(25,91)(26,94)(27,89)(28,92)(29,95)(30,90)(31,93)(32,96)(33,81)(34,84)(35,87)(36,82)(37,85)(38,88)(39,83)(40,86)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(65,71)(67,69)(68,72)(74,76)(75,79)(78,80)>;
G:=Group( (1,42,19,77,70,54)(2,43,20,78,71,55)(3,44,21,79,72,56)(4,45,22,80,65,49)(5,46,23,73,66,50)(6,47,24,74,67,51)(7,48,17,75,68,52)(8,41,18,76,69,53)(9,96,83,63,28,35)(10,89,84,64,29,36)(11,90,85,57,30,37)(12,91,86,58,31,38)(13,92,87,59,32,39)(14,93,88,60,25,40)(15,94,81,61,26,33)(16,95,82,62,27,34), (1,37)(2,86)(3,39)(4,88)(5,33)(6,82)(7,35)(8,84)(9,52)(10,18)(11,54)(12,20)(13,56)(14,22)(15,50)(16,24)(17,63)(19,57)(21,59)(23,61)(25,65)(26,46)(27,67)(28,48)(29,69)(30,42)(31,71)(32,44)(34,74)(36,76)(38,78)(40,80)(41,89)(43,91)(45,93)(47,95)(49,60)(51,62)(53,64)(55,58)(66,94)(68,96)(70,90)(72,92)(73,81)(75,83)(77,85)(79,87), (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), (2,4)(3,7)(6,8)(9,59)(10,62)(11,57)(12,60)(13,63)(14,58)(15,61)(16,64)(17,21)(18,24)(20,22)(25,91)(26,94)(27,89)(28,92)(29,95)(30,90)(31,93)(32,96)(33,81)(34,84)(35,87)(36,82)(37,85)(38,88)(39,83)(40,86)(41,47)(43,45)(44,48)(49,55)(51,53)(52,56)(65,71)(67,69)(68,72)(74,76)(75,79)(78,80) );
G=PermutationGroup([[(1,42,19,77,70,54),(2,43,20,78,71,55),(3,44,21,79,72,56),(4,45,22,80,65,49),(5,46,23,73,66,50),(6,47,24,74,67,51),(7,48,17,75,68,52),(8,41,18,76,69,53),(9,96,83,63,28,35),(10,89,84,64,29,36),(11,90,85,57,30,37),(12,91,86,58,31,38),(13,92,87,59,32,39),(14,93,88,60,25,40),(15,94,81,61,26,33),(16,95,82,62,27,34)], [(1,37),(2,86),(3,39),(4,88),(5,33),(6,82),(7,35),(8,84),(9,52),(10,18),(11,54),(12,20),(13,56),(14,22),(15,50),(16,24),(17,63),(19,57),(21,59),(23,61),(25,65),(26,46),(27,67),(28,48),(29,69),(30,42),(31,71),(32,44),(34,74),(36,76),(38,78),(40,80),(41,89),(43,91),(45,93),(47,95),(49,60),(51,62),(53,64),(55,58),(66,94),(68,96),(70,90),(72,92),(73,81),(75,83),(77,85),(79,87)], [(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)], [(2,4),(3,7),(6,8),(9,59),(10,62),(11,57),(12,60),(13,63),(14,58),(15,61),(16,64),(17,21),(18,24),(20,22),(25,91),(26,94),(27,89),(28,92),(29,95),(30,90),(31,93),(32,96),(33,81),(34,84),(35,87),(36,82),(37,85),(38,88),(39,83),(40,86),(41,47),(43,45),(44,48),(49,55),(51,53),(52,56),(65,71),(67,69),(68,72),(74,76),(75,79),(78,80)]])
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 | 2 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D4 | D6 | D6 | D6 | SD16 | C3⋊D4 | C8.C22 | S3×D4 | S3×D4 | S3×SD16 | D4.D6 |
| kernel | D6⋊8SD16 | C2.Dic12 | D6⋊C8 | Q8⋊2Dic3 | C2×D4.S3 | D6⋊3D4 | C6×SD16 | C2×S3×Q8 | C2×SD16 | Dic6 | C2×Dic3 | C3×Q8 | C22×S3 | C2×C8 | C2×D4 | 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⋊8SD16 ►in GL4(𝔽73) generated by
| 9 | 0 | 0 | 0 |
| 53 | 65 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 55 | 65 | 0 | 0 |
| 13 | 18 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 72 | 0 | 0 | 0 |
| 41 | 1 | 0 | 0 |
| 0 | 0 | 67 | 6 |
| 0 | 0 | 67 | 67 |
| 1 | 0 | 0 | 0 |
| 32 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [9,53,0,0,0,65,0,0,0,0,1,0,0,0,0,1],[55,13,0,0,65,18,0,0,0,0,1,0,0,0,0,1],[72,41,0,0,0,1,0,0,0,0,67,67,0,0,6,67],[1,32,0,0,0,72,0,0,0,0,1,0,0,0,0,72] >;
D6⋊8SD16 in GAP, Magma, Sage, TeX
D_6\rtimes_8{\rm SD}_{16} % in TeX
G:=Group("D6:8SD16"); // GroupNames label
G:=SmallGroup(192,729);
// by ID
G=gap.SmallGroup(192,729);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,232,254,219,851,438,102,6278]);
// 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=c^3>;
// generators/relations