metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D6.3C42, C42.183D6, Dic3⋊5M4(2), Dic3.3C42, C8⋊9(C4×S3), C8⋊S3⋊3C4, C24⋊20(C2×C4), C8⋊C4⋊12S3, C3⋊2(C4×M4(2)), (C2×C8).270D6, C6.9(C2×C42), (C8×Dic3)⋊27C2, C2.10(S3×C42), C2.2(S3×M4(2)), (S3×C42).15C2, (C4×Dic3).16C4, C6.18(C2×M4(2)), (C4×C12).228C22, C42.S3⋊19C2, (C2×C24).270C22, (C2×C12).813C23, C12.126(C22×C4), (C4×Dic3).299C22, C3⋊C8⋊18(C2×C4), (S3×C2×C4).16C4, C4.100(S3×C2×C4), (C3×C8⋊C4)⋊13C2, C22.40(S3×C2×C4), (C4×S3).20(C2×C4), (C2×C4).128(C4×S3), (C2×C8⋊S3).11C2, (C2×C12).147(C2×C4), (C2×C3⋊C8).296C22, (S3×C2×C4).270C22, (C2×C6).68(C22×C4), (C22×S3).53(C2×C4), (C2×C4).755(C22×S3), (C2×Dic3).82(C2×C4), SmallGroup(192,266)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic3⋊5M4(2)
G = < a,b,c,d | a6=c8=d2=1, b2=a3, bab-1=cac-1=dad=a-1, bc=cb, bd=db, dcd=c5 >
Subgroups: 280 in 142 conjugacy classes, 79 normal (19 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, C2×C4, C23, Dic3, C12, C12, D6, D6, C2×C6, C42, C42, C2×C8, C2×C8, M4(2), C22×C4, C3⋊C8, C24, C4×S3, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C4×C8, C8⋊C4, C8⋊C4, C2×C42, C2×M4(2), C8⋊S3, C2×C3⋊C8, C4×Dic3, C4×Dic3, C4×C12, C2×C24, S3×C2×C4, S3×C2×C4, C4×M4(2), C42.S3, C8×Dic3, C3×C8⋊C4, S3×C42, C2×C8⋊S3, Dic3⋊5M4(2)
Quotients: C1, C2, C4, C22, S3, C2×C4, C23, D6, C42, M4(2), C22×C4, C4×S3, C22×S3, C2×C42, C2×M4(2), S3×C2×C4, C4×M4(2), S3×C42, S3×M4(2), Dic3⋊5M4(2)
►On 96 points
(1 62 41 23 74 51)(2 52 75 24 42 63)(3 64 43 17 76 53)(4 54 77 18 44 57)(5 58 45 19 78 55)(6 56 79 20 46 59)(7 60 47 21 80 49)(8 50 73 22 48 61)(9 40 91 69 84 31)(10 32 85 70 92 33)(11 34 93 71 86 25)(12 26 87 72 94 35)(13 36 95 65 88 27)(14 28 81 66 96 37)(15 38 89 67 82 29)(16 30 83 68 90 39)
(1 66 23 14)(2 67 24 15)(3 68 17 16)(4 69 18 9)(5 70 19 10)(6 71 20 11)(7 72 21 12)(8 65 22 13)(25 56 93 46)(26 49 94 47)(27 50 95 48)(28 51 96 41)(29 52 89 42)(30 53 90 43)(31 54 91 44)(32 55 92 45)(33 58 85 78)(34 59 86 79)(35 60 87 80)(36 61 88 73)(37 62 81 74)(38 63 82 75)(39 64 83 76)(40 57 84 77)
(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 5)(3 7)(10 14)(12 16)(17 21)(19 23)(25 34)(26 39)(27 36)(28 33)(29 38)(30 35)(31 40)(32 37)(41 78)(42 75)(43 80)(44 77)(45 74)(46 79)(47 76)(48 73)(49 64)(50 61)(51 58)(52 63)(53 60)(54 57)(55 62)(56 59)(66 70)(68 72)(81 92)(82 89)(83 94)(84 91)(85 96)(86 93)(87 90)(88 95)
G:=sub<Sym(96)| (1,62,41,23,74,51)(2,52,75,24,42,63)(3,64,43,17,76,53)(4,54,77,18,44,57)(5,58,45,19,78,55)(6,56,79,20,46,59)(7,60,47,21,80,49)(8,50,73,22,48,61)(9,40,91,69,84,31)(10,32,85,70,92,33)(11,34,93,71,86,25)(12,26,87,72,94,35)(13,36,95,65,88,27)(14,28,81,66,96,37)(15,38,89,67,82,29)(16,30,83,68,90,39), (1,66,23,14)(2,67,24,15)(3,68,17,16)(4,69,18,9)(5,70,19,10)(6,71,20,11)(7,72,21,12)(8,65,22,13)(25,56,93,46)(26,49,94,47)(27,50,95,48)(28,51,96,41)(29,52,89,42)(30,53,90,43)(31,54,91,44)(32,55,92,45)(33,58,85,78)(34,59,86,79)(35,60,87,80)(36,61,88,73)(37,62,81,74)(38,63,82,75)(39,64,83,76)(40,57,84,77), (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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,34)(26,39)(27,36)(28,33)(29,38)(30,35)(31,40)(32,37)(41,78)(42,75)(43,80)(44,77)(45,74)(46,79)(47,76)(48,73)(49,64)(50,61)(51,58)(52,63)(53,60)(54,57)(55,62)(56,59)(66,70)(68,72)(81,92)(82,89)(83,94)(84,91)(85,96)(86,93)(87,90)(88,95)>;
G:=Group( (1,62,41,23,74,51)(2,52,75,24,42,63)(3,64,43,17,76,53)(4,54,77,18,44,57)(5,58,45,19,78,55)(6,56,79,20,46,59)(7,60,47,21,80,49)(8,50,73,22,48,61)(9,40,91,69,84,31)(10,32,85,70,92,33)(11,34,93,71,86,25)(12,26,87,72,94,35)(13,36,95,65,88,27)(14,28,81,66,96,37)(15,38,89,67,82,29)(16,30,83,68,90,39), (1,66,23,14)(2,67,24,15)(3,68,17,16)(4,69,18,9)(5,70,19,10)(6,71,20,11)(7,72,21,12)(8,65,22,13)(25,56,93,46)(26,49,94,47)(27,50,95,48)(28,51,96,41)(29,52,89,42)(30,53,90,43)(31,54,91,44)(32,55,92,45)(33,58,85,78)(34,59,86,79)(35,60,87,80)(36,61,88,73)(37,62,81,74)(38,63,82,75)(39,64,83,76)(40,57,84,77), (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), (1,5)(3,7)(10,14)(12,16)(17,21)(19,23)(25,34)(26,39)(27,36)(28,33)(29,38)(30,35)(31,40)(32,37)(41,78)(42,75)(43,80)(44,77)(45,74)(46,79)(47,76)(48,73)(49,64)(50,61)(51,58)(52,63)(53,60)(54,57)(55,62)(56,59)(66,70)(68,72)(81,92)(82,89)(83,94)(84,91)(85,96)(86,93)(87,90)(88,95) );
G=PermutationGroup([[(1,62,41,23,74,51),(2,52,75,24,42,63),(3,64,43,17,76,53),(4,54,77,18,44,57),(5,58,45,19,78,55),(6,56,79,20,46,59),(7,60,47,21,80,49),(8,50,73,22,48,61),(9,40,91,69,84,31),(10,32,85,70,92,33),(11,34,93,71,86,25),(12,26,87,72,94,35),(13,36,95,65,88,27),(14,28,81,66,96,37),(15,38,89,67,82,29),(16,30,83,68,90,39)], [(1,66,23,14),(2,67,24,15),(3,68,17,16),(4,69,18,9),(5,70,19,10),(6,71,20,11),(7,72,21,12),(8,65,22,13),(25,56,93,46),(26,49,94,47),(27,50,95,48),(28,51,96,41),(29,52,89,42),(30,53,90,43),(31,54,91,44),(32,55,92,45),(33,58,85,78),(34,59,86,79),(35,60,87,80),(36,61,88,73),(37,62,81,74),(38,63,82,75),(39,64,83,76),(40,57,84,77)], [(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)], [(1,5),(3,7),(10,14),(12,16),(17,21),(19,23),(25,34),(26,39),(27,36),(28,33),(29,38),(30,35),(31,40),(32,37),(41,78),(42,75),(43,80),(44,77),(45,74),(46,79),(47,76),(48,73),(49,64),(50,61),(51,58),(52,63),(53,60),(54,57),(55,62),(56,59),(66,70),(68,72),(81,92),(82,89),(83,94),(84,91),(85,96),(86,93),(87,90),(88,95)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | ··· | 4P | 4Q | 4R | 6A | 6B | 6C | 8A | ··· | 8H | 8I | ··· | 8P | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | ··· | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | ··· | 3 | 6 | 6 | 2 | 2 | 2 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | + | |||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C4 | S3 | D6 | D6 | M4(2) | C4×S3 | C4×S3 | S3×M4(2) |
| kernel | Dic3⋊5M4(2) | C42.S3 | C8×Dic3 | C3×C8⋊C4 | S3×C42 | C2×C8⋊S3 | C8⋊S3 | C4×Dic3 | S3×C2×C4 | C8⋊C4 | C42 | C2×C8 | Dic3 | C8 | C2×C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 1 | 2 | 16 | 4 | 4 | 1 | 1 | 2 | 8 | 8 | 4 | 4 |
Matrix representation of Dic3⋊5M4(2) ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 1 | 0 |
| 46 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 |
| 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 27 |
| 0 | 72 | 0 | 0 |
| 27 | 0 | 0 | 0 |
| 0 | 0 | 46 | 27 |
| 0 | 0 | 0 | 27 |
| 72 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 72 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,1,1,0,0,72,0],[46,0,0,0,0,46,0,0,0,0,46,0,0,0,27,27],[0,27,0,0,72,0,0,0,0,0,46,0,0,0,27,27],[72,0,0,0,0,1,0,0,0,0,1,0,0,0,72,72] >;
Dic3⋊5M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_3\rtimes_5M_4(2) % in TeX
G:=Group("Dic3:5M4(2)"); // GroupNames label
G:=SmallGroup(192,266);
// by ID
G=gap.SmallGroup(192,266);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,120,758,387,58,136,6278]);
// 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=d*a*d=a^-1,b*c=c*b,b*d=d*b,d*c*d=c^5>;
// generators/relations