metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3⋊1M4(2), C3⋊C8⋊16D4, D6⋊C8⋊25C2, C3⋊2(C8⋊6D4), D6⋊C4.6C4, C6.28(C4×D4), C22⋊C8⋊14S3, C4.198(S3×D4), (C2×C8).196D6, Dic3⋊C8⋊25C2, (C8×Dic3)⋊16C2, C6.10(C8○D4), C12.357(C2×D4), Dic3⋊C4.6C4, C23.19(C4×S3), (C22×C4).96D6, C2.12(C8○D12), C6.D4.6C4, C2.14(S3×M4(2)), C6.23(C2×M4(2)), C12.299(C4○D4), (C2×C12).824C23, (C2×C24).216C22, C4.125(D4⋊2S3), (C22×C12).95C22, C2.12(Dic3⋊4D4), (C4×Dic3).272C22, (C2×C4).33(C4×S3), (C4×C3⋊D4).2C2, (C2×C3⋊D4).6C4, (C2×C8⋊S3)⋊13C2, (C3×C22⋊C8)⋊23C2, (C2×C12).41(C2×C4), (C2×C4.Dic3)⋊2C2, C22.106(S3×C2×C4), (C2×C3⋊C8).192C22, (S3×C2×C4).179C22, (C2×C6).79(C22×C4), (C22×C6).42(C2×C4), (C22×S3).13(C2×C4), (C2×C4).766(C22×S3), (C2×Dic3).51(C2×C4), SmallGroup(192,288)
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=cac-1=a-1, ad=da, cbc-1=dbd=a3b, dcd=c5 >
Subgroups: 288 in 122 conjugacy classes, 51 normal (47 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, Dic3, C12, C12, D6, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), C22×C4, C22×C4, C2×D4, C3⋊C8, C3⋊C8, C24, C4×S3, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C4×D4, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4.Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C6.D4, C2×C24, S3×C2×C4, C2×C3⋊D4, C22×C12, C8⋊6D4, C8×Dic3, Dic3⋊C8, D6⋊C8, C3×C22⋊C8, C2×C8⋊S3, C2×C4.Dic3, C4×C3⋊D4, Dic3⋊M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, M4(2), C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C2×M4(2), C8○D4, S3×C2×C4, S3×D4, D4⋊2S3, C8⋊6D4, Dic3⋊4D4, C8○D12, S3×M4(2), Dic3⋊M4(2)
►On 96 points
(1 57 73 87 38 13)(2 14 39 88 74 58)(3 59 75 81 40 15)(4 16 33 82 76 60)(5 61 77 83 34 9)(6 10 35 84 78 62)(7 63 79 85 36 11)(8 12 37 86 80 64)(17 48 29 92 52 66)(18 67 53 93 30 41)(19 42 31 94 54 68)(20 69 55 95 32 43)(21 44 25 96 56 70)(22 71 49 89 26 45)(23 46 27 90 50 72)(24 65 51 91 28 47)
(1 72 87 27)(2 28 88 65)(3 66 81 29)(4 30 82 67)(5 68 83 31)(6 32 84 69)(7 70 85 25)(8 26 86 71)(9 19 77 94)(10 95 78 20)(11 21 79 96)(12 89 80 22)(13 23 73 90)(14 91 74 24)(15 17 75 92)(16 93 76 18)(33 53 60 41)(34 42 61 54)(35 55 62 43)(36 44 63 56)(37 49 64 45)(38 46 57 50)(39 51 58 47)(40 48 59 52)
(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 68)(2 65)(3 70)(4 67)(5 72)(6 69)(7 66)(8 71)(9 50)(10 55)(11 52)(12 49)(13 54)(14 51)(15 56)(16 53)(17 63)(18 60)(19 57)(20 62)(21 59)(22 64)(23 61)(24 58)(25 81)(26 86)(27 83)(28 88)(29 85)(30 82)(31 87)(32 84)(33 93)(34 90)(35 95)(36 92)(37 89)(38 94)(39 91)(40 96)(41 76)(42 73)(43 78)(44 75)(45 80)(46 77)(47 74)(48 79)
G:=sub<Sym(96)| (1,57,73,87,38,13)(2,14,39,88,74,58)(3,59,75,81,40,15)(4,16,33,82,76,60)(5,61,77,83,34,9)(6,10,35,84,78,62)(7,63,79,85,36,11)(8,12,37,86,80,64)(17,48,29,92,52,66)(18,67,53,93,30,41)(19,42,31,94,54,68)(20,69,55,95,32,43)(21,44,25,96,56,70)(22,71,49,89,26,45)(23,46,27,90,50,72)(24,65,51,91,28,47), (1,72,87,27)(2,28,88,65)(3,66,81,29)(4,30,82,67)(5,68,83,31)(6,32,84,69)(7,70,85,25)(8,26,86,71)(9,19,77,94)(10,95,78,20)(11,21,79,96)(12,89,80,22)(13,23,73,90)(14,91,74,24)(15,17,75,92)(16,93,76,18)(33,53,60,41)(34,42,61,54)(35,55,62,43)(36,44,63,56)(37,49,64,45)(38,46,57,50)(39,51,58,47)(40,48,59,52), (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), (1,68)(2,65)(3,70)(4,67)(5,72)(6,69)(7,66)(8,71)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(25,81)(26,86)(27,83)(28,88)(29,85)(30,82)(31,87)(32,84)(33,93)(34,90)(35,95)(36,92)(37,89)(38,94)(39,91)(40,96)(41,76)(42,73)(43,78)(44,75)(45,80)(46,77)(47,74)(48,79)>;
G:=Group( (1,57,73,87,38,13)(2,14,39,88,74,58)(3,59,75,81,40,15)(4,16,33,82,76,60)(5,61,77,83,34,9)(6,10,35,84,78,62)(7,63,79,85,36,11)(8,12,37,86,80,64)(17,48,29,92,52,66)(18,67,53,93,30,41)(19,42,31,94,54,68)(20,69,55,95,32,43)(21,44,25,96,56,70)(22,71,49,89,26,45)(23,46,27,90,50,72)(24,65,51,91,28,47), (1,72,87,27)(2,28,88,65)(3,66,81,29)(4,30,82,67)(5,68,83,31)(6,32,84,69)(7,70,85,25)(8,26,86,71)(9,19,77,94)(10,95,78,20)(11,21,79,96)(12,89,80,22)(13,23,73,90)(14,91,74,24)(15,17,75,92)(16,93,76,18)(33,53,60,41)(34,42,61,54)(35,55,62,43)(36,44,63,56)(37,49,64,45)(38,46,57,50)(39,51,58,47)(40,48,59,52), (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), (1,68)(2,65)(3,70)(4,67)(5,72)(6,69)(7,66)(8,71)(9,50)(10,55)(11,52)(12,49)(13,54)(14,51)(15,56)(16,53)(17,63)(18,60)(19,57)(20,62)(21,59)(22,64)(23,61)(24,58)(25,81)(26,86)(27,83)(28,88)(29,85)(30,82)(31,87)(32,84)(33,93)(34,90)(35,95)(36,92)(37,89)(38,94)(39,91)(40,96)(41,76)(42,73)(43,78)(44,75)(45,80)(46,77)(47,74)(48,79) );
G=PermutationGroup([[(1,57,73,87,38,13),(2,14,39,88,74,58),(3,59,75,81,40,15),(4,16,33,82,76,60),(5,61,77,83,34,9),(6,10,35,84,78,62),(7,63,79,85,36,11),(8,12,37,86,80,64),(17,48,29,92,52,66),(18,67,53,93,30,41),(19,42,31,94,54,68),(20,69,55,95,32,43),(21,44,25,96,56,70),(22,71,49,89,26,45),(23,46,27,90,50,72),(24,65,51,91,28,47)], [(1,72,87,27),(2,28,88,65),(3,66,81,29),(4,30,82,67),(5,68,83,31),(6,32,84,69),(7,70,85,25),(8,26,86,71),(9,19,77,94),(10,95,78,20),(11,21,79,96),(12,89,80,22),(13,23,73,90),(14,91,74,24),(15,17,75,92),(16,93,76,18),(33,53,60,41),(34,42,61,54),(35,55,62,43),(36,44,63,56),(37,49,64,45),(38,46,57,50),(39,51,58,47),(40,48,59,52)], [(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)], [(1,68),(2,65),(3,70),(4,67),(5,72),(6,69),(7,66),(8,71),(9,50),(10,55),(11,52),(12,49),(13,54),(14,51),(15,56),(16,53),(17,63),(18,60),(19,57),(20,62),(21,59),(22,64),(23,61),(24,58),(25,81),(26,86),(27,83),(28,88),(29,85),(30,82),(31,87),(32,84),(33,93),(34,90),(35,95),(36,92),(37,89),(38,94),(39,91),(40,96),(41,76),(42,73),(43,78),(44,75),(45,80),(46,77),(47,74),(48,79)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 12 | 2 | 1 | 1 | 1 | 1 | 4 | 6 | 6 | 6 | 6 | 12 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | |||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | C4 | S3 | D4 | D6 | D6 | M4(2) | C4○D4 | C4×S3 | C4×S3 | C8○D4 | C8○D12 | S3×D4 | D4⋊2S3 | S3×M4(2) |
| kernel | Dic3⋊M4(2) | C8×Dic3 | Dic3⋊C8 | D6⋊C8 | C3×C22⋊C8 | C2×C8⋊S3 | C2×C4.Dic3 | C4×C3⋊D4 | Dic3⋊C4 | D6⋊C4 | C6.D4 | C2×C3⋊D4 | C22⋊C8 | C3⋊C8 | C2×C8 | C22×C4 | Dic3 | C12 | C2×C4 | C23 | C6 | C2 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 2 | 2 | 1 | 4 | 2 | 2 | 2 | 4 | 8 | 1 | 1 | 2 |
Matrix representation of Dic3⋊M4(2) ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 72 | 3 | 0 | 0 | 0 | 0 |
| 48 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 25 | 0 | 0 |
| 0 | 0 | 70 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 22 | 0 | 0 | 0 | 0 | 0 |
| 39 | 51 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 72 | 3 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 48 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,72],[72,48,0,0,0,0,3,1,0,0,0,0,0,0,72,70,0,0,0,0,25,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72],[22,39,0,0,0,0,0,51,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,46,27,0,0,0,0,0,27],[72,0,0,0,0,0,3,1,0,0,0,0,0,0,1,0,0,0,0,0,48,72,0,0,0,0,0,0,1,0,0,0,0,0,0,1] >;
Dic3⋊M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes M_4(2) % in TeX
G:=Group("Dic3:M4(2)"); // GroupNames label
G:=SmallGroup(192,288);
// by ID
G=gap.SmallGroup(192,288);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,701,422,219,58,136,6278]);
// Polycyclic
G:=Group<a,b,c,d|a^6=c^8=d^2=1,b^2=a^3,b*a*b^-1=c*a*c^-1=a^-1,a*d=d*a,c*b*c^-1=d*b*d=a^3*b,d*c*d=c^5>;
// generators/relations