metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.4SD16, C4.Q8:7S3, D6:C8:30C2, C4:C4.37D6, (C2xC8).138D6, C12:D4.4C2, C2.D24:31C2, C6.D8:14C2, C2.23(S3xSD16), C6.39(C2xSD16), C12.28(C4oD4), C4.73(C4oD12), C2.21(Q8:3D6), C6.69(C8:C22), C12.Q8:17C2, (C2xDic3).41D4, (C22xS3).83D4, C22.215(S3xD4), (C2xC24).285C22, (C2xC12).279C23, C4.25(Q8:3S3), (C2xD12).73C22, C3:3(C23.46D4), C2.12(D6.D4), C4:Dic3.111C22, C6.42(C22.D4), (S3xC4:C4):6C2, (C3xC4.Q8):16C2, (C2xC6).284(C2xD4), (C2xC3:C8).57C22, (S3xC2xC4).32C22, (C3xC4:C4).72C22, (C2xC4).382(C22xS3), SmallGroup(192,422)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for D6.4SD16
G = < a,b,c,d | a6=b2=c8=1, d2=a3, bab=a-1, ac=ca, ad=da, cbc-1=dbd-1=a3b, dcd-1=c3 >
Subgroups: 400 in 114 conjugacy classes, 39 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C2xC4, C2xC4, D4, C23, Dic3, C12, C12, D6, D6, C2xC6, C22:C4, C4:C4, C4:C4, C2xC8, C2xC8, C22xC4, C2xD4, C3:C8, C24, C4xS3, D12, C2xDic3, C2xDic3, C2xC12, C2xC12, C22xS3, C22xS3, C22:C8, D4:C4, C4.Q8, C4.Q8, C2xC4:C4, C4:D4, C2xC3:C8, Dic3:C4, C4:Dic3, D6:C4, C3xC4:C4, C2xC24, S3xC2xC4, S3xC2xC4, C2xD12, C2xD12, C23.46D4, C12.Q8, C6.D8, D6:C8, C2.D24, C3xC4.Q8, S3xC4:C4, C12:D4, D6.4SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2xD4, C4oD4, C22xS3, C22.D4, C2xSD16, C8:C22, C4oD12, S3xD4, Q8:3S3, C23.46D4, D6.D4, S3xSD16, Q8:3D6, D6.4SD16
►On 96 points
(1 72 88 17 77 30)(2 65 81 18 78 31)(3 66 82 19 79 32)(4 67 83 20 80 25)(5 68 84 21 73 26)(6 69 85 22 74 27)(7 70 86 23 75 28)(8 71 87 24 76 29)(9 36 61 49 94 48)(10 37 62 50 95 41)(11 38 63 51 96 42)(12 39 64 52 89 43)(13 40 57 53 90 44)(14 33 58 54 91 45)(15 34 59 55 92 46)(16 35 60 56 93 47)
(1 35)(2 94)(3 37)(4 96)(5 39)(6 90)(7 33)(8 92)(9 78)(10 66)(11 80)(12 68)(13 74)(14 70)(15 76)(16 72)(17 93)(18 36)(19 95)(20 38)(21 89)(22 40)(23 91)(24 34)(25 42)(26 64)(27 44)(28 58)(29 46)(30 60)(31 48)(32 62)(41 82)(43 84)(45 86)(47 88)(49 65)(50 79)(51 67)(52 73)(53 69)(54 75)(55 71)(56 77)(57 85)(59 87)(61 81)(63 83)
(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 45 17 58)(2 48 18 61)(3 43 19 64)(4 46 20 59)(5 41 21 62)(6 44 22 57)(7 47 23 60)(8 42 24 63)(9 78 49 65)(10 73 50 68)(11 76 51 71)(12 79 52 66)(13 74 53 69)(14 77 54 72)(15 80 55 67)(16 75 56 70)(25 92 83 34)(26 95 84 37)(27 90 85 40)(28 93 86 35)(29 96 87 38)(30 91 88 33)(31 94 81 36)(32 89 82 39)
G:=sub<Sym(96)| (1,72,88,17,77,30)(2,65,81,18,78,31)(3,66,82,19,79,32)(4,67,83,20,80,25)(5,68,84,21,73,26)(6,69,85,22,74,27)(7,70,86,23,75,28)(8,71,87,24,76,29)(9,36,61,49,94,48)(10,37,62,50,95,41)(11,38,63,51,96,42)(12,39,64,52,89,43)(13,40,57,53,90,44)(14,33,58,54,91,45)(15,34,59,55,92,46)(16,35,60,56,93,47), (1,35)(2,94)(3,37)(4,96)(5,39)(6,90)(7,33)(8,92)(9,78)(10,66)(11,80)(12,68)(13,74)(14,70)(15,76)(16,72)(17,93)(18,36)(19,95)(20,38)(21,89)(22,40)(23,91)(24,34)(25,42)(26,64)(27,44)(28,58)(29,46)(30,60)(31,48)(32,62)(41,82)(43,84)(45,86)(47,88)(49,65)(50,79)(51,67)(52,73)(53,69)(54,75)(55,71)(56,77)(57,85)(59,87)(61,81)(63,83), (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), (1,45,17,58)(2,48,18,61)(3,43,19,64)(4,46,20,59)(5,41,21,62)(6,44,22,57)(7,47,23,60)(8,42,24,63)(9,78,49,65)(10,73,50,68)(11,76,51,71)(12,79,52,66)(13,74,53,69)(14,77,54,72)(15,80,55,67)(16,75,56,70)(25,92,83,34)(26,95,84,37)(27,90,85,40)(28,93,86,35)(29,96,87,38)(30,91,88,33)(31,94,81,36)(32,89,82,39)>;
G:=Group( (1,72,88,17,77,30)(2,65,81,18,78,31)(3,66,82,19,79,32)(4,67,83,20,80,25)(5,68,84,21,73,26)(6,69,85,22,74,27)(7,70,86,23,75,28)(8,71,87,24,76,29)(9,36,61,49,94,48)(10,37,62,50,95,41)(11,38,63,51,96,42)(12,39,64,52,89,43)(13,40,57,53,90,44)(14,33,58,54,91,45)(15,34,59,55,92,46)(16,35,60,56,93,47), (1,35)(2,94)(3,37)(4,96)(5,39)(6,90)(7,33)(8,92)(9,78)(10,66)(11,80)(12,68)(13,74)(14,70)(15,76)(16,72)(17,93)(18,36)(19,95)(20,38)(21,89)(22,40)(23,91)(24,34)(25,42)(26,64)(27,44)(28,58)(29,46)(30,60)(31,48)(32,62)(41,82)(43,84)(45,86)(47,88)(49,65)(50,79)(51,67)(52,73)(53,69)(54,75)(55,71)(56,77)(57,85)(59,87)(61,81)(63,83), (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), (1,45,17,58)(2,48,18,61)(3,43,19,64)(4,46,20,59)(5,41,21,62)(6,44,22,57)(7,47,23,60)(8,42,24,63)(9,78,49,65)(10,73,50,68)(11,76,51,71)(12,79,52,66)(13,74,53,69)(14,77,54,72)(15,80,55,67)(16,75,56,70)(25,92,83,34)(26,95,84,37)(27,90,85,40)(28,93,86,35)(29,96,87,38)(30,91,88,33)(31,94,81,36)(32,89,82,39) );
G=PermutationGroup([[(1,72,88,17,77,30),(2,65,81,18,78,31),(3,66,82,19,79,32),(4,67,83,20,80,25),(5,68,84,21,73,26),(6,69,85,22,74,27),(7,70,86,23,75,28),(8,71,87,24,76,29),(9,36,61,49,94,48),(10,37,62,50,95,41),(11,38,63,51,96,42),(12,39,64,52,89,43),(13,40,57,53,90,44),(14,33,58,54,91,45),(15,34,59,55,92,46),(16,35,60,56,93,47)], [(1,35),(2,94),(3,37),(4,96),(5,39),(6,90),(7,33),(8,92),(9,78),(10,66),(11,80),(12,68),(13,74),(14,70),(15,76),(16,72),(17,93),(18,36),(19,95),(20,38),(21,89),(22,40),(23,91),(24,34),(25,42),(26,64),(27,44),(28,58),(29,46),(30,60),(31,48),(32,62),(41,82),(43,84),(45,86),(47,88),(49,65),(50,79),(51,67),(52,73),(53,69),(54,75),(55,71),(56,77),(57,85),(59,87),(61,81),(63,83)], [(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)], [(1,45,17,58),(2,48,18,61),(3,43,19,64),(4,46,20,59),(5,41,21,62),(6,44,22,57),(7,47,23,60),(8,42,24,63),(9,78,49,65),(10,73,50,68),(11,76,51,71),(12,79,52,66),(13,74,53,69),(14,77,54,72),(15,80,55,67),(16,75,56,70),(25,92,83,34),(26,95,84,37),(27,90,85,40),(28,93,86,35),(29,96,87,38),(30,91,88,33),(31,94,81,36),(32,89,82,39)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 24 | 2 | 2 | 2 | 4 | 4 | 8 | 12 | 12 | 12 | 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 | C4oD4 | SD16 | C4oD12 | C8:C22 | Q8:3S3 | S3xD4 | S3xSD16 | Q8:3D6 |
| kernel | D6.4SD16 | C12.Q8 | C6.D8 | D6:C8 | C2.D24 | C3xC4.Q8 | S3xC4:C4 | C12:D4 | C4.Q8 | C2xDic3 | C22xS3 | C4:C4 | C2xC8 | 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 D6.4SD16 ►in GL4(F73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 72 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 7 | 7 |
| 0 | 0 | 14 | 66 |
| 67 | 67 | 0 | 0 |
| 6 | 67 | 0 | 0 |
| 0 | 0 | 30 | 13 |
| 0 | 0 | 60 | 43 |
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 7 | 59 |
| 0 | 0 | 14 | 66 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,1,0,0,0,0,0,72,0,0,1,1],[1,0,0,0,0,1,0,0,0,0,7,14,0,0,7,66],[67,6,0,0,67,67,0,0,0,0,30,60,0,0,13,43],[1,0,0,0,0,72,0,0,0,0,7,14,0,0,59,66] >;
D6.4SD16 in GAP, Magma, Sage, TeX
D_6._4{\rm SD}_{16} % in TeX
G:=Group("D6.4SD16"); // GroupNames label
G:=SmallGroup(192,422);
// by ID
G=gap.SmallGroup(192,422);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,64,926,219,100,851,102,6278]);
// 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=d*b*d^-1=a^3*b,d*c*d^-1=c^3>;
// generators/relations