metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.1SD16, D6⋊C8.5C2, C4⋊C4.151D6, C8⋊Dic3⋊14C2, (C2×C8).122D6, (C2×Q8).39D6, Q8⋊C4⋊10S3, Q8⋊2Dic3⋊7C2, D6⋊3Q8.1C2, C6.30(C2×SD16), C2.16(S3×SD16), C4.55(C4○D12), C12.Q8⋊10C2, (C2×Dic3).29D4, (C22×S3).76D4, C22.195(S3×D4), (C6×Q8).28C22, C12.161(C4○D4), C4.86(D4⋊2S3), (C2×C12).245C23, (C2×C24).133C22, C2.13(Q16⋊S3), C6.59(C8.C22), C3⋊2(C23.47D4), C4⋊Dic3.93C22, C2.16(C23.9D6), C6.24(C22.D4), (S3×C4⋊C4).2C2, (C2×C6).258(C2×D4), (C2×C3⋊C8).37C22, (S3×C2×C4).19C22, (C3×Q8⋊C4)⋊10C2, (C3×C4⋊C4).46C22, (C2×C4).352(C22×S3), SmallGroup(192,364)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.1SD16
G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd-1=a3c3 >
Subgroups: 296 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, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×Q8, C22×S3, C22⋊C8, Q8⋊C4, Q8⋊C4, C4.Q8, C2×C4⋊C4, C22⋊Q8, C2×C3⋊C8, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, S3×C2×C4, S3×C2×C4, C6×Q8, C23.47D4, C12.Q8, C8⋊Dic3, D6⋊C8, Q8⋊2Dic3, C3×Q8⋊C4, S3×C4⋊C4, D6⋊3Q8, D6.1SD16
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, C23.47D4, C23.9D6, S3×SD16, Q16⋊S3, D6.1SD16
►On 96 points
Generators in S96
(1 24 58 29 80 85)(2 17 59 30 73 86)(3 18 60 31 74 87)(4 19 61 32 75 88)(5 20 62 25 76 81)(6 21 63 26 77 82)(7 22 64 27 78 83)(8 23 57 28 79 84)(9 40 47 49 72 92)(10 33 48 50 65 93)(11 34 41 51 66 94)(12 35 42 52 67 95)(13 36 43 53 68 96)(14 37 44 54 69 89)(15 38 45 55 70 90)(16 39 46 56 71 91)
(1 81)(2 63)(3 83)(4 57)(5 85)(6 59)(7 87)(8 61)(9 96)(10 44)(11 90)(12 46)(13 92)(14 48)(15 94)(16 42)(17 21)(18 78)(19 23)(20 80)(22 74)(24 76)(25 58)(26 86)(27 60)(28 88)(29 62)(30 82)(31 64)(32 84)(33 37)(34 70)(35 39)(36 72)(38 66)(40 68)(41 55)(43 49)(45 51)(47 53)(50 89)(52 91)(54 93)(56 95)(65 69)(67 71)(73 77)(75 79)
(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 11 29 51)(2 54 30 14)(3 9 31 49)(4 52 32 12)(5 15 25 55)(6 50 26 10)(7 13 27 53)(8 56 28 16)(17 69 73 37)(18 40 74 72)(19 67 75 35)(20 38 76 70)(21 65 77 33)(22 36 78 68)(23 71 79 39)(24 34 80 66)(41 85 94 58)(42 61 95 88)(43 83 96 64)(44 59 89 86)(45 81 90 62)(46 57 91 84)(47 87 92 60)(48 63 93 82)
G:=sub<Sym(96)| (1,24,58,29,80,85)(2,17,59,30,73,86)(3,18,60,31,74,87)(4,19,61,32,75,88)(5,20,62,25,76,81)(6,21,63,26,77,82)(7,22,64,27,78,83)(8,23,57,28,79,84)(9,40,47,49,72,92)(10,33,48,50,65,93)(11,34,41,51,66,94)(12,35,42,52,67,95)(13,36,43,53,68,96)(14,37,44,54,69,89)(15,38,45,55,70,90)(16,39,46,56,71,91), (1,81)(2,63)(3,83)(4,57)(5,85)(6,59)(7,87)(8,61)(9,96)(10,44)(11,90)(12,46)(13,92)(14,48)(15,94)(16,42)(17,21)(18,78)(19,23)(20,80)(22,74)(24,76)(25,58)(26,86)(27,60)(28,88)(29,62)(30,82)(31,64)(32,84)(33,37)(34,70)(35,39)(36,72)(38,66)(40,68)(41,55)(43,49)(45,51)(47,53)(50,89)(52,91)(54,93)(56,95)(65,69)(67,71)(73,77)(75,79), (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), (1,11,29,51)(2,54,30,14)(3,9,31,49)(4,52,32,12)(5,15,25,55)(6,50,26,10)(7,13,27,53)(8,56,28,16)(17,69,73,37)(18,40,74,72)(19,67,75,35)(20,38,76,70)(21,65,77,33)(22,36,78,68)(23,71,79,39)(24,34,80,66)(41,85,94,58)(42,61,95,88)(43,83,96,64)(44,59,89,86)(45,81,90,62)(46,57,91,84)(47,87,92,60)(48,63,93,82)>;
G:=Group( (1,24,58,29,80,85)(2,17,59,30,73,86)(3,18,60,31,74,87)(4,19,61,32,75,88)(5,20,62,25,76,81)(6,21,63,26,77,82)(7,22,64,27,78,83)(8,23,57,28,79,84)(9,40,47,49,72,92)(10,33,48,50,65,93)(11,34,41,51,66,94)(12,35,42,52,67,95)(13,36,43,53,68,96)(14,37,44,54,69,89)(15,38,45,55,70,90)(16,39,46,56,71,91), (1,81)(2,63)(3,83)(4,57)(5,85)(6,59)(7,87)(8,61)(9,96)(10,44)(11,90)(12,46)(13,92)(14,48)(15,94)(16,42)(17,21)(18,78)(19,23)(20,80)(22,74)(24,76)(25,58)(26,86)(27,60)(28,88)(29,62)(30,82)(31,64)(32,84)(33,37)(34,70)(35,39)(36,72)(38,66)(40,68)(41,55)(43,49)(45,51)(47,53)(50,89)(52,91)(54,93)(56,95)(65,69)(67,71)(73,77)(75,79), (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), (1,11,29,51)(2,54,30,14)(3,9,31,49)(4,52,32,12)(5,15,25,55)(6,50,26,10)(7,13,27,53)(8,56,28,16)(17,69,73,37)(18,40,74,72)(19,67,75,35)(20,38,76,70)(21,65,77,33)(22,36,78,68)(23,71,79,39)(24,34,80,66)(41,85,94,58)(42,61,95,88)(43,83,96,64)(44,59,89,86)(45,81,90,62)(46,57,91,84)(47,87,92,60)(48,63,93,82) );
G=PermutationGroup([[(1,24,58,29,80,85),(2,17,59,30,73,86),(3,18,60,31,74,87),(4,19,61,32,75,88),(5,20,62,25,76,81),(6,21,63,26,77,82),(7,22,64,27,78,83),(8,23,57,28,79,84),(9,40,47,49,72,92),(10,33,48,50,65,93),(11,34,41,51,66,94),(12,35,42,52,67,95),(13,36,43,53,68,96),(14,37,44,54,69,89),(15,38,45,55,70,90),(16,39,46,56,71,91)], [(1,81),(2,63),(3,83),(4,57),(5,85),(6,59),(7,87),(8,61),(9,96),(10,44),(11,90),(12,46),(13,92),(14,48),(15,94),(16,42),(17,21),(18,78),(19,23),(20,80),(22,74),(24,76),(25,58),(26,86),(27,60),(28,88),(29,62),(30,82),(31,64),(32,84),(33,37),(34,70),(35,39),(36,72),(38,66),(40,68),(41,55),(43,49),(45,51),(47,53),(50,89),(52,91),(54,93),(56,95),(65,69),(67,71),(73,77),(75,79)], [(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)], [(1,11,29,51),(2,54,30,14),(3,9,31,49),(4,52,32,12),(5,15,25,55),(6,50,26,10),(7,13,27,53),(8,56,28,16),(17,69,73,37),(18,40,74,72),(19,67,75,35),(20,38,76,70),(21,65,77,33),(22,36,78,68),(23,71,79,39),(24,34,80,66),(41,85,94,58),(42,61,95,88),(43,83,96,64),(44,59,89,86),(45,81,90,62),(46,57,91,84),(47,87,92,60),(48,63,93,82)]])
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 | 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 | S3×SD16 | Q16⋊S3 |
| kernel | D6.1SD16 | C12.Q8 | C8⋊Dic3 | D6⋊C8 | Q8⋊2Dic3 | C3×Q8⋊C4 | S3×C4⋊C4 | D6⋊3Q8 | Q8⋊C4 | C2×Dic3 | C22×S3 | C4⋊C4 | C2×C8 | C2×Q8 | 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 D6.1SD16 ►in GL6(𝔽73)
| 72 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 72 | 72 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 25 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 65 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 61 |
| 0 | 0 | 0 | 0 | 6 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 46 | 0 | 0 | 0 |
| 0 | 0 | 0 | 46 | 0 | 0 |
| 0 | 0 | 0 | 0 | 58 | 34 |
| 0 | 0 | 0 | 0 | 2 | 15 |
G:=sub<GL(6,GF(73))| [72,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[72,0,0,0,0,0,72,1,0,0,0,0,0,0,72,25,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,65,27,0,0,0,0,0,0,12,6,0,0,0,0,61,0],[72,0,0,0,0,0,0,72,0,0,0,0,0,0,46,0,0,0,0,0,0,46,0,0,0,0,0,0,58,2,0,0,0,0,34,15] >;
D6.1SD16 in GAP, Magma, Sage, TeX
D_6._1{\rm SD}_{16} % in TeX
G:=Group("D6.1SD16"); // GroupNames label
G:=SmallGroup(192,364);
// by ID
G=gap.SmallGroup(192,364);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,254,219,184,851,438,102,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=b^2=c^8=1,d^2=a^3,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=a^3*c^3>;
// generators/relations