metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.4M4(2), C24⋊C4⋊11C2, (C2×C8).162D6, C22⋊C8.7S3, Dic3⋊C8⋊21C2, C23.50(C4×S3), (C2×C6).7M4(2), C3⋊2(C42.6C4), (C4×Dic3).18C4, (C22×C4).317D6, C6.20(C2×M4(2)), C2.11(S3×M4(2)), C12.295(C4○D4), (C2×C12).817C23, (C2×C24).211C22, C22.7(C8⋊S3), C4.121(D4⋊2S3), (C22×Dic3).8C4, C6.21(C42⋊C2), C12.55D4.14C2, (C22×C12).335C22, (C4×Dic3).269C22, C2.9(C23.16D6), C2.8(C2×C8⋊S3), (C2×C4).130(C4×S3), C22.101(S3×C2×C4), (C2×C4×Dic3).30C2, (C2×C12).151(C2×C4), (C3×C22⋊C8).13C2, (C2×C3⋊C8).190C22, (C2×C6).72(C22×C4), (C22×C6).35(C2×C4), (C2×C4).759(C22×S3), (C2×Dic3).83(C2×C4), SmallGroup(192,278)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3.M4(2)
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=a3c5 >
Subgroups: 224 in 110 conjugacy classes, 53 normal (33 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C22, C6, C6, C8, C2×C4, C2×C4, C23, Dic3, Dic3, C12, C12, C2×C6, C2×C6, C2×C6, C42, C2×C8, C2×C8, C22×C4, C22×C4, C3⋊C8, C24, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×C6, C8⋊C4, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C2×C24, C22×Dic3, C22×C12, C42.6C4, Dic3⋊C8, C24⋊C4, C12.55D4, C3×C22⋊C8, C2×C4×Dic3, Dic3.M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C2×M4(2), C8⋊S3, S3×C2×C4, D4⋊2S3, C42.6C4, C23.16D6, C2×C8⋊S3, S3×M4(2), Dic3.M4(2)
►On 96 points
(1 39 10 83 77 58)(2 40 11 84 78 59)(3 33 12 85 79 60)(4 34 13 86 80 61)(5 35 14 87 73 62)(6 36 15 88 74 63)(7 37 16 81 75 64)(8 38 9 82 76 57)(17 43 29 91 71 55)(18 44 30 92 72 56)(19 45 31 93 65 49)(20 46 32 94 66 50)(21 47 25 95 67 51)(22 48 26 96 68 52)(23 41 27 89 69 53)(24 42 28 90 70 54)
(1 55 83 29)(2 30 84 56)(3 49 85 31)(4 32 86 50)(5 51 87 25)(6 26 88 52)(7 53 81 27)(8 28 82 54)(9 24 57 90)(10 91 58 17)(11 18 59 92)(12 93 60 19)(13 20 61 94)(14 95 62 21)(15 22 63 96)(16 89 64 23)(33 65 79 45)(34 46 80 66)(35 67 73 47)(36 48 74 68)(37 69 75 41)(38 42 76 70)(39 71 77 43)(40 44 78 72)
(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 88)(4 82)(6 84)(8 86)(9 61)(11 63)(13 57)(15 59)(18 96)(20 90)(22 92)(24 94)(26 56)(28 50)(30 52)(32 54)(34 76)(36 78)(38 80)(40 74)(42 66)(44 68)(46 70)(48 72)
G:=sub<Sym(96)| (1,39,10,83,77,58)(2,40,11,84,78,59)(3,33,12,85,79,60)(4,34,13,86,80,61)(5,35,14,87,73,62)(6,36,15,88,74,63)(7,37,16,81,75,64)(8,38,9,82,76,57)(17,43,29,91,71,55)(18,44,30,92,72,56)(19,45,31,93,65,49)(20,46,32,94,66,50)(21,47,25,95,67,51)(22,48,26,96,68,52)(23,41,27,89,69,53)(24,42,28,90,70,54), (1,55,83,29)(2,30,84,56)(3,49,85,31)(4,32,86,50)(5,51,87,25)(6,26,88,52)(7,53,81,27)(8,28,82,54)(9,24,57,90)(10,91,58,17)(11,18,59,92)(12,93,60,19)(13,20,61,94)(14,95,62,21)(15,22,63,96)(16,89,64,23)(33,65,79,45)(34,46,80,66)(35,67,73,47)(36,48,74,68)(37,69,75,41)(38,42,76,70)(39,71,77,43)(40,44,78,72), (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), (2,88)(4,82)(6,84)(8,86)(9,61)(11,63)(13,57)(15,59)(18,96)(20,90)(22,92)(24,94)(26,56)(28,50)(30,52)(32,54)(34,76)(36,78)(38,80)(40,74)(42,66)(44,68)(46,70)(48,72)>;
G:=Group( (1,39,10,83,77,58)(2,40,11,84,78,59)(3,33,12,85,79,60)(4,34,13,86,80,61)(5,35,14,87,73,62)(6,36,15,88,74,63)(7,37,16,81,75,64)(8,38,9,82,76,57)(17,43,29,91,71,55)(18,44,30,92,72,56)(19,45,31,93,65,49)(20,46,32,94,66,50)(21,47,25,95,67,51)(22,48,26,96,68,52)(23,41,27,89,69,53)(24,42,28,90,70,54), (1,55,83,29)(2,30,84,56)(3,49,85,31)(4,32,86,50)(5,51,87,25)(6,26,88,52)(7,53,81,27)(8,28,82,54)(9,24,57,90)(10,91,58,17)(11,18,59,92)(12,93,60,19)(13,20,61,94)(14,95,62,21)(15,22,63,96)(16,89,64,23)(33,65,79,45)(34,46,80,66)(35,67,73,47)(36,48,74,68)(37,69,75,41)(38,42,76,70)(39,71,77,43)(40,44,78,72), (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), (2,88)(4,82)(6,84)(8,86)(9,61)(11,63)(13,57)(15,59)(18,96)(20,90)(22,92)(24,94)(26,56)(28,50)(30,52)(32,54)(34,76)(36,78)(38,80)(40,74)(42,66)(44,68)(46,70)(48,72) );
G=PermutationGroup([[(1,39,10,83,77,58),(2,40,11,84,78,59),(3,33,12,85,79,60),(4,34,13,86,80,61),(5,35,14,87,73,62),(6,36,15,88,74,63),(7,37,16,81,75,64),(8,38,9,82,76,57),(17,43,29,91,71,55),(18,44,30,92,72,56),(19,45,31,93,65,49),(20,46,32,94,66,50),(21,47,25,95,67,51),(22,48,26,96,68,52),(23,41,27,89,69,53),(24,42,28,90,70,54)], [(1,55,83,29),(2,30,84,56),(3,49,85,31),(4,32,86,50),(5,51,87,25),(6,26,88,52),(7,53,81,27),(8,28,82,54),(9,24,57,90),(10,91,58,17),(11,18,59,92),(12,93,60,19),(13,20,61,94),(14,95,62,21),(15,22,63,96),(16,89,64,23),(33,65,79,45),(34,46,80,66),(35,67,73,47),(36,48,74,68),(37,69,75,41),(38,42,76,70),(39,71,77,43),(40,44,78,72)], [(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)], [(2,88),(4,82),(6,84),(8,86),(9,61),(11,63),(13,57),(15,59),(18,96),(20,90),(22,92),(24,94),(26,56),(28,50),(30,52),(32,54),(34,76),(36,78),(38,80),(40,74),(42,66),(44,68),(46,70),(48,72)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 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 | 1 | 1 | 1 | 1 | 2 | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4○D4 | M4(2) | C4×S3 | C4×S3 | C8⋊S3 | D4⋊2S3 | S3×M4(2) |
| kernel | Dic3.M4(2) | Dic3⋊C8 | C24⋊C4 | C12.55D4 | C3×C22⋊C8 | C2×C4×Dic3 | C4×Dic3 | C22×Dic3 | C22⋊C8 | C2×C8 | C22×C4 | Dic3 | C12 | C2×C6 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 1 | 2 | 1 | 4 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of Dic3.M4(2) ►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 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 25 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 23 | 55 |
| 0 | 0 | 0 | 0 | 5 | 50 |
| 46 | 3 | 0 | 0 | 0 | 0 |
| 25 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 49 | 3 | 0 | 0 |
| 0 | 0 | 18 | 24 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 18 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 16 | 72 | 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],[46,25,0,0,0,0,0,27,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,23,5,0,0,0,0,55,50],[46,25,0,0,0,0,3,27,0,0,0,0,0,0,49,18,0,0,0,0,3,24,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,18,0,0,0,0,0,72,0,0,0,0,0,0,1,16,0,0,0,0,0,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3.M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_3.M_4(2) % in TeX
G:=Group("Dic3.M4(2)"); // GroupNames label
G:=SmallGroup(192,278);
// by ID
G=gap.SmallGroup(192,278);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,758,219,58,136,6278]);
// 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=a^3*c^5>;
// generators/relations