metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊3SD16, (C3×D4).7D4, (C2×C8).143D6, (C2×Q8).74D6, Dic3⋊C8⋊32C2, (C2×D4).143D6, C12.170(C2×D4), C3⋊5(D4.D4), D4.2(C3⋊D4), (C2×SD16).3S3, (C6×SD16).6C2, (D4×Dic3).8C2, C2.26(S3×SD16), C6.43(C2×SD16), Dic3⋊Q8⋊3C2, C12.96(C4○D4), C4.9(D4⋊2S3), Q8⋊2Dic3⋊25C2, C2.Dic12⋊33C2, (C6×D4).88C22, C22.260(S3×D4), (C6×Q8).69C22, C6.112(C4⋊D4), (C2×C24).290C22, (C2×C12).439C23, (C2×Dic3).182D4, C2.26(D4.D6), C6.45(C8.C22), C4⋊Dic3.169C22, (C4×Dic3).46C22, C2.24(C23.14D6), (C2×Dic6).123C22, C4.38(C2×C3⋊D4), (C2×C6).351(C2×D4), (C2×D4.S3).8C2, (C2×C3⋊C8).151C22, (C2×C4).528(C22×S3), SmallGroup(192,721)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊3SD16
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=a-1, ac=ca, ad=da, cbc-1=a3b, bd=db, dcd=c3 >
Subgroups: 328 in 120 conjugacy classes, 43 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C3×Q8, C22×C6, Q8⋊C4, C4⋊C8, C4×D4, C4⋊Q8, C2×SD16, C2×SD16, C2×C3⋊C8, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D4.S3, C6.D4, C2×C24, C3×SD16, C2×Dic6, C22×Dic3, C6×D4, C6×Q8, D4.D4, Dic3⋊C8, C2.Dic12, Q8⋊2Dic3, C2×D4.S3, D4×Dic3, Dic3⋊Q8, C6×SD16, Dic3⋊3SD16
Quotients: C1, C2, C22, S3, D4, C23, D6, SD16, C2×D4, C4○D4, C3⋊D4, C22×S3, C4⋊D4, C2×SD16, C8.C22, S3×D4, D4⋊2S3, C2×C3⋊D4, D4.D4, S3×SD16, D4.D6, C23.14D6, Dic3⋊3SD16
►On 96 points
(1 53 69 19 43 77)(2 54 70 20 44 78)(3 55 71 21 45 79)(4 56 72 22 46 80)(5 49 65 23 47 73)(6 50 66 24 48 74)(7 51 67 17 41 75)(8 52 68 18 42 76)(9 26 95 62 81 36)(10 27 96 63 82 37)(11 28 89 64 83 38)(12 29 90 57 84 39)(13 30 91 58 85 40)(14 31 92 59 86 33)(15 32 93 60 87 34)(16 25 94 61 88 35)
(1 12 19 57)(2 58 20 13)(3 14 21 59)(4 60 22 15)(5 16 23 61)(6 62 24 9)(7 10 17 63)(8 64 18 11)(25 65 88 73)(26 74 81 66)(27 67 82 75)(28 76 83 68)(29 69 84 77)(30 78 85 70)(31 71 86 79)(32 80 87 72)(33 45 92 55)(34 56 93 46)(35 47 94 49)(36 50 95 48)(37 41 96 51)(38 52 89 42)(39 43 90 53)(40 54 91 44)
(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 11)(10 14)(13 15)(17 21)(18 24)(20 22)(26 28)(27 31)(30 32)(33 37)(34 40)(36 38)(41 45)(42 48)(44 46)(50 52)(51 55)(54 56)(58 60)(59 63)(62 64)(66 68)(67 71)(70 72)(74 76)(75 79)(78 80)(81 83)(82 86)(85 87)(89 95)(91 93)(92 96)
G:=sub<Sym(96)| (1,53,69,19,43,77)(2,54,70,20,44,78)(3,55,71,21,45,79)(4,56,72,22,46,80)(5,49,65,23,47,73)(6,50,66,24,48,74)(7,51,67,17,41,75)(8,52,68,18,42,76)(9,26,95,62,81,36)(10,27,96,63,82,37)(11,28,89,64,83,38)(12,29,90,57,84,39)(13,30,91,58,85,40)(14,31,92,59,86,33)(15,32,93,60,87,34)(16,25,94,61,88,35), (1,12,19,57)(2,58,20,13)(3,14,21,59)(4,60,22,15)(5,16,23,61)(6,62,24,9)(7,10,17,63)(8,64,18,11)(25,65,88,73)(26,74,81,66)(27,67,82,75)(28,76,83,68)(29,69,84,77)(30,78,85,70)(31,71,86,79)(32,80,87,72)(33,45,92,55)(34,56,93,46)(35,47,94,49)(36,50,95,48)(37,41,96,51)(38,52,89,42)(39,43,90,53)(40,54,91,44), (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)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96)>;
G:=Group( (1,53,69,19,43,77)(2,54,70,20,44,78)(3,55,71,21,45,79)(4,56,72,22,46,80)(5,49,65,23,47,73)(6,50,66,24,48,74)(7,51,67,17,41,75)(8,52,68,18,42,76)(9,26,95,62,81,36)(10,27,96,63,82,37)(11,28,89,64,83,38)(12,29,90,57,84,39)(13,30,91,58,85,40)(14,31,92,59,86,33)(15,32,93,60,87,34)(16,25,94,61,88,35), (1,12,19,57)(2,58,20,13)(3,14,21,59)(4,60,22,15)(5,16,23,61)(6,62,24,9)(7,10,17,63)(8,64,18,11)(25,65,88,73)(26,74,81,66)(27,67,82,75)(28,76,83,68)(29,69,84,77)(30,78,85,70)(31,71,86,79)(32,80,87,72)(33,45,92,55)(34,56,93,46)(35,47,94,49)(36,50,95,48)(37,41,96,51)(38,52,89,42)(39,43,90,53)(40,54,91,44), (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)(9,11)(10,14)(13,15)(17,21)(18,24)(20,22)(26,28)(27,31)(30,32)(33,37)(34,40)(36,38)(41,45)(42,48)(44,46)(50,52)(51,55)(54,56)(58,60)(59,63)(62,64)(66,68)(67,71)(70,72)(74,76)(75,79)(78,80)(81,83)(82,86)(85,87)(89,95)(91,93)(92,96) );
G=PermutationGroup([[(1,53,69,19,43,77),(2,54,70,20,44,78),(3,55,71,21,45,79),(4,56,72,22,46,80),(5,49,65,23,47,73),(6,50,66,24,48,74),(7,51,67,17,41,75),(8,52,68,18,42,76),(9,26,95,62,81,36),(10,27,96,63,82,37),(11,28,89,64,83,38),(12,29,90,57,84,39),(13,30,91,58,85,40),(14,31,92,59,86,33),(15,32,93,60,87,34),(16,25,94,61,88,35)], [(1,12,19,57),(2,58,20,13),(3,14,21,59),(4,60,22,15),(5,16,23,61),(6,62,24,9),(7,10,17,63),(8,64,18,11),(25,65,88,73),(26,74,81,66),(27,67,82,75),(28,76,83,68),(29,69,84,77),(30,78,85,70),(31,71,86,79),(32,80,87,72),(33,45,92,55),(34,56,93,46),(35,47,94,49),(36,50,95,48),(37,41,96,51),(38,52,89,42),(39,43,90,53),(40,54,91,44)], [(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),(9,11),(10,14),(13,15),(17,21),(18,24),(20,22),(26,28),(27,31),(30,32),(33,37),(34,40),(36,38),(41,45),(42,48),(44,46),(50,52),(51,55),(54,56),(58,60),(59,63),(62,64),(66,68),(67,71),(70,72),(74,76),(75,79),(78,80),(81,83),(82,86),(85,87),(89,95),(91,93),(92,96)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 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 | 4 | 4 | 2 | 2 | 2 | 6 | 6 | 8 | 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 | SD16 | C4○D4 | C3⋊D4 | C8.C22 | D4⋊2S3 | S3×D4 | S3×SD16 | D4.D6 |
| kernel | Dic3⋊3SD16 | Dic3⋊C8 | C2.Dic12 | Q8⋊2Dic3 | C2×D4.S3 | D4×Dic3 | Dic3⋊Q8 | C6×SD16 | C2×SD16 | C2×Dic3 | C3×D4 | C2×C8 | C2×D4 | C2×Q8 | Dic3 | C12 | D4 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of Dic3⋊3SD16 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 7 | 59 | 0 | 0 | 0 | 0 |
| 14 | 66 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 51 | 32 |
| 0 | 0 | 0 | 0 | 10 | 22 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 67 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 71 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[7,14,0,0,0,0,59,66,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,51,10,0,0,0,0,32,22],[0,1,0,0,0,0,1,0,0,0,0,0,0,0,12,67,0,0,0,0,12,0,0,0,0,0,0,0,72,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,71,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3⋊3SD16 in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_3{\rm SD}_{16} % in TeX
G:=Group("Dic3:3SD16"); // GroupNames label
G:=SmallGroup(192,721);
// by ID
G=gap.SmallGroup(192,721);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,422,135,184,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=a^-1,a*c=c*a,a*d=d*a,c*b*c^-1=a^3*b,b*d=d*b,d*c*d=c^3>;
// generators/relations