metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: M4(2)⋊2Dic3, C12.9(C4⋊C4), (C2×C12).9Q8, C4.Dic3⋊5C4, (C2×C6).3C42, (C2×C4).4Dic6, C4.34(D6⋊C4), (C2×C12).468D4, (C2×C4).128D12, (C3×M4(2))⋊5C4, C23.48(C4×S3), C4.3(C4⋊Dic3), (C22×C4).77D6, C6.5(C4.D4), (C2×M4(2)).5S3, (C6×M4(2)).9C2, C22.3(C4×Dic3), C12.86(C22⋊C4), C3⋊2(C22.C42), C4.24(Dic3⋊C4), C22.41(D6⋊C4), C6.5(C4.10D4), (C22×Dic3).3C4, C2.2(C12.47D4), C2.2(C12.46D4), C4.17(C6.D4), C22.5(Dic3⋊C4), C2.13(C6.C42), C6.13(C2.C42), (C22×C12).123C22, (C2×C6).6(C4⋊C4), (C2×C4).20(C4×S3), (C2×C12).61(C2×C4), (C2×C4).21(C3⋊D4), (C2×C4⋊Dic3).28C2, (C22×C6).31(C2×C4), (C2×C4).14(C2×Dic3), (C2×C6).53(C22⋊C4), (C2×C4.Dic3).10C2, SmallGroup(192,113)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for M4(2)⋊Dic3
G = < a,b,c,d | a8=b2=c6=1, d2=c3, bab=a5, ac=ca, dad-1=ab, bc=cb, bd=db, dcd-1=c-1 >
Subgroups: 232 in 98 conjugacy classes, 51 normal (43 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, M4(2), M4(2), C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×C12, C22×C6, C2×C4⋊C4, C2×M4(2), C2×M4(2), C2×C3⋊C8, C4.Dic3, C4.Dic3, C4⋊Dic3, C2×C24, C3×M4(2), C3×M4(2), C22×Dic3, C22×C12, C22.C42, C2×C4.Dic3, C2×C4⋊Dic3, C6×M4(2), M4(2)⋊Dic3
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Q8, Dic3, D6, C42, C22⋊C4, C4⋊C4, Dic6, C4×S3, D12, C2×Dic3, C3⋊D4, C2.C42, C4.D4, C4.10D4, C4×Dic3, Dic3⋊C4, C4⋊Dic3, D6⋊C4, C6.D4, C22.C42, C12.46D4, C12.47D4, C6.C42, M4(2)⋊Dic3
►On 96 points
(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 63)(2 60)(3 57)(4 62)(5 59)(6 64)(7 61)(8 58)(9 90)(10 95)(11 92)(12 89)(13 94)(14 91)(15 96)(16 93)(17 73)(18 78)(19 75)(20 80)(21 77)(22 74)(23 79)(24 76)(25 45)(26 42)(27 47)(28 44)(29 41)(30 46)(31 43)(32 48)(33 68)(34 65)(35 70)(36 67)(37 72)(38 69)(39 66)(40 71)(49 86)(50 83)(51 88)(52 85)(53 82)(54 87)(55 84)(56 81)
(1 54 77 63 87 21)(2 55 78 64 88 22)(3 56 79 57 81 23)(4 49 80 58 82 24)(5 50 73 59 83 17)(6 51 74 60 84 18)(7 52 75 61 85 19)(8 53 76 62 86 20)(9 40 31 94 67 47)(10 33 32 95 68 48)(11 34 25 96 69 41)(12 35 26 89 70 42)(13 36 27 90 71 43)(14 37 28 91 72 44)(15 38 29 92 65 45)(16 39 30 93 66 46)
(1 89 63 12)(2 9 64 94)(3 95 57 10)(4 15 58 92)(5 93 59 16)(6 13 60 90)(7 91 61 14)(8 11 62 96)(17 66 73 39)(18 36 74 71)(19 72 75 37)(20 34 76 69)(21 70 77 35)(22 40 78 67)(23 68 79 33)(24 38 80 65)(25 53 41 86)(26 87 42 54)(27 51 43 84)(28 85 44 52)(29 49 45 82)(30 83 46 50)(31 55 47 88)(32 81 48 56)
G:=sub<Sym(96)| (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), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,45)(26,42)(27,47)(28,44)(29,41)(30,46)(31,43)(32,48)(33,68)(34,65)(35,70)(36,67)(37,72)(38,69)(39,66)(40,71)(49,86)(50,83)(51,88)(52,85)(53,82)(54,87)(55,84)(56,81), (1,54,77,63,87,21)(2,55,78,64,88,22)(3,56,79,57,81,23)(4,49,80,58,82,24)(5,50,73,59,83,17)(6,51,74,60,84,18)(7,52,75,61,85,19)(8,53,76,62,86,20)(9,40,31,94,67,47)(10,33,32,95,68,48)(11,34,25,96,69,41)(12,35,26,89,70,42)(13,36,27,90,71,43)(14,37,28,91,72,44)(15,38,29,92,65,45)(16,39,30,93,66,46), (1,89,63,12)(2,9,64,94)(3,95,57,10)(4,15,58,92)(5,93,59,16)(6,13,60,90)(7,91,61,14)(8,11,62,96)(17,66,73,39)(18,36,74,71)(19,72,75,37)(20,34,76,69)(21,70,77,35)(22,40,78,67)(23,68,79,33)(24,38,80,65)(25,53,41,86)(26,87,42,54)(27,51,43,84)(28,85,44,52)(29,49,45,82)(30,83,46,50)(31,55,47,88)(32,81,48,56)>;
G:=Group( (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), (1,63)(2,60)(3,57)(4,62)(5,59)(6,64)(7,61)(8,58)(9,90)(10,95)(11,92)(12,89)(13,94)(14,91)(15,96)(16,93)(17,73)(18,78)(19,75)(20,80)(21,77)(22,74)(23,79)(24,76)(25,45)(26,42)(27,47)(28,44)(29,41)(30,46)(31,43)(32,48)(33,68)(34,65)(35,70)(36,67)(37,72)(38,69)(39,66)(40,71)(49,86)(50,83)(51,88)(52,85)(53,82)(54,87)(55,84)(56,81), (1,54,77,63,87,21)(2,55,78,64,88,22)(3,56,79,57,81,23)(4,49,80,58,82,24)(5,50,73,59,83,17)(6,51,74,60,84,18)(7,52,75,61,85,19)(8,53,76,62,86,20)(9,40,31,94,67,47)(10,33,32,95,68,48)(11,34,25,96,69,41)(12,35,26,89,70,42)(13,36,27,90,71,43)(14,37,28,91,72,44)(15,38,29,92,65,45)(16,39,30,93,66,46), (1,89,63,12)(2,9,64,94)(3,95,57,10)(4,15,58,92)(5,93,59,16)(6,13,60,90)(7,91,61,14)(8,11,62,96)(17,66,73,39)(18,36,74,71)(19,72,75,37)(20,34,76,69)(21,70,77,35)(22,40,78,67)(23,68,79,33)(24,38,80,65)(25,53,41,86)(26,87,42,54)(27,51,43,84)(28,85,44,52)(29,49,45,82)(30,83,46,50)(31,55,47,88)(32,81,48,56) );
G=PermutationGroup([[(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)], [(1,63),(2,60),(3,57),(4,62),(5,59),(6,64),(7,61),(8,58),(9,90),(10,95),(11,92),(12,89),(13,94),(14,91),(15,96),(16,93),(17,73),(18,78),(19,75),(20,80),(21,77),(22,74),(23,79),(24,76),(25,45),(26,42),(27,47),(28,44),(29,41),(30,46),(31,43),(32,48),(33,68),(34,65),(35,70),(36,67),(37,72),(38,69),(39,66),(40,71),(49,86),(50,83),(51,88),(52,85),(53,82),(54,87),(55,84),(56,81)], [(1,54,77,63,87,21),(2,55,78,64,88,22),(3,56,79,57,81,23),(4,49,80,58,82,24),(5,50,73,59,83,17),(6,51,74,60,84,18),(7,52,75,61,85,19),(8,53,76,62,86,20),(9,40,31,94,67,47),(10,33,32,95,68,48),(11,34,25,96,69,41),(12,35,26,89,70,42),(13,36,27,90,71,43),(14,37,28,91,72,44),(15,38,29,92,65,45),(16,39,30,93,66,46)], [(1,89,63,12),(2,9,64,94),(3,95,57,10),(4,15,58,92),(5,93,59,16),(6,13,60,90),(7,91,61,14),(8,11,62,96),(17,66,73,39),(18,36,74,71),(19,72,75,37),(20,34,76,69),(21,70,77,35),(22,40,78,67),(23,68,79,33),(24,38,80,65),(25,53,41,86),(26,87,42,54),(27,51,43,84),(28,85,44,52),(29,49,45,82),(30,83,46,50),(31,55,47,88),(32,81,48,56)]])
42 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
42 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | - | + | - | + | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D4 | Q8 | Dic3 | D6 | Dic6 | C4×S3 | D12 | C3⋊D4 | C4×S3 | C4.D4 | C4.10D4 | C12.46D4 | C12.47D4 |
| kernel | M4(2)⋊Dic3 | C2×C4.Dic3 | C2×C4⋊Dic3 | C6×M4(2) | C4.Dic3 | C3×M4(2) | C22×Dic3 | C2×M4(2) | C2×C12 | C2×C12 | M4(2) | C22×C4 | C2×C4 | C2×C4 | C2×C4 | C2×C4 | C23 | C6 | C6 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 4 | 4 | 4 | 1 | 3 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 1 | 1 | 2 | 2 |
Matrix representation of M4(2)⋊Dic3 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 13 | 25 | 25 |
| 0 | 0 | 60 | 30 | 23 | 25 |
| 0 | 0 | 62 | 63 | 43 | 60 |
| 0 | 0 | 72 | 62 | 13 | 30 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 17 | 17 | 72 | 0 |
| 0 | 0 | 0 | 17 | 0 | 72 |
| 1 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 1 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 23 | 0 | 0 |
| 0 | 0 | 18 | 68 | 0 | 0 |
| 0 | 0 | 26 | 14 | 50 | 5 |
| 0 | 0 | 14 | 61 | 55 | 23 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,43,60,62,72,0,0,13,30,63,62,0,0,25,23,43,13,0,0,25,25,60,30],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,17,0,0,0,0,1,17,17,0,0,0,0,72,0,0,0,0,0,0,72],[1,1,0,0,0,0,72,0,0,0,0,0,0,0,0,72,0,0,0,0,1,72,0,0,0,0,0,0,72,72,0,0,0,0,1,0],[0,27,0,0,0,0,27,0,0,0,0,0,0,0,5,18,26,14,0,0,23,68,14,61,0,0,0,0,50,55,0,0,0,0,5,23] >;
M4(2)⋊Dic3 in GAP, Magma, Sage, TeX
M_4(2)\rtimes {\rm Dic}_3 % in TeX
G:=Group("M4(2):Dic3"); // GroupNames label
G:=SmallGroup(192,113);
// by ID
G=gap.SmallGroup(192,113);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,253,64,1123,136,851,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^8=b^2=c^6=1,d^2=c^3,b*a*b=a^5,a*c=c*a,d*a*d^-1=a*b,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations