metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: Dic3.5M4(2), (C2×Dic3)⋊3C8, (C2×C8).192D6, C22⋊C8.9S3, C6.5(C22×C8), C22.6(S3×C8), Dic3⋊C8⋊20C2, (C8×Dic3)⋊13C2, C23.49(C4×S3), Dic3.8(C2×C8), C2.3(S3×M4(2)), (C4×Dic3).17C4, (C22×C4).316D6, C6.19(C2×M4(2)), C12.294(C4○D4), (C2×C12).816C23, (C2×C24).210C22, C3⋊2(C42.12C4), C4.120(D4⋊2S3), (C22×Dic3).7C4, C6.20(C42⋊C2), C12.55D4.13C2, (C22×C12).334C22, (C4×Dic3).300C22, C2.3(C23.16D6), C2.7(S3×C2×C8), (C2×C6).4(C2×C8), C22.42(S3×C2×C4), (C2×C4).129(C4×S3), (C2×C4×Dic3).29C2, (C2×C12).150(C2×C4), (C3×C22⋊C8).12C2, (C2×C3⋊C8).298C22, (C2×C6).71(C22×C4), (C22×C6).34(C2×C4), (C2×C4).758(C22×S3), (C2×Dic3).108(C2×C4), SmallGroup(192,277)
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=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd=a3c5 >
Subgroups: 224 in 118 conjugacy classes, 61 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×Dic3, C2×C12, C2×C12, C22×C6, C4×C8, C22⋊C8, C22⋊C8, C4⋊C8, C2×C42, C2×C3⋊C8, C4×Dic3, C2×C24, C22×Dic3, C22×C12, C42.12C4, C8×Dic3, Dic3⋊C8, C12.55D4, C3×C22⋊C8, C2×C4×Dic3, Dic3.5M4(2)
Quotients: C1, C2, C4, C22, S3, C8, C2×C4, C23, D6, C2×C8, M4(2), C22×C4, C4○D4, C4×S3, C22×S3, C42⋊C2, C22×C8, C2×M4(2), S3×C8, S3×C2×C4, D4⋊2S3, C42.12C4, C23.16D6, S3×C2×C8, S3×M4(2), Dic3.5M4(2)
►On 96 points
(1 39 15 83 50 61)(2 40 16 84 51 62)(3 33 9 85 52 63)(4 34 10 86 53 64)(5 35 11 87 54 57)(6 36 12 88 55 58)(7 37 13 81 56 59)(8 38 14 82 49 60)(17 76 90 65 25 48)(18 77 91 66 26 41)(19 78 92 67 27 42)(20 79 93 68 28 43)(21 80 94 69 29 44)(22 73 95 70 30 45)(23 74 96 71 31 46)(24 75 89 72 32 47)
(1 72 83 24)(2 65 84 17)(3 66 85 18)(4 67 86 19)(5 68 87 20)(6 69 88 21)(7 70 81 22)(8 71 82 23)(9 77 63 26)(10 78 64 27)(11 79 57 28)(12 80 58 29)(13 73 59 30)(14 74 60 31)(15 75 61 32)(16 76 62 25)(33 91 52 41)(34 92 53 42)(35 93 54 43)(36 94 55 44)(37 95 56 45)(38 96 49 46)(39 89 50 47)(40 90 51 48)
(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)(10 60)(12 62)(14 64)(16 58)(17 69)(19 71)(21 65)(23 67)(25 80)(27 74)(29 76)(31 78)(34 49)(36 51)(38 53)(40 55)(42 96)(44 90)(46 92)(48 94)
G:=sub<Sym(96)| (1,39,15,83,50,61)(2,40,16,84,51,62)(3,33,9,85,52,63)(4,34,10,86,53,64)(5,35,11,87,54,57)(6,36,12,88,55,58)(7,37,13,81,56,59)(8,38,14,82,49,60)(17,76,90,65,25,48)(18,77,91,66,26,41)(19,78,92,67,27,42)(20,79,93,68,28,43)(21,80,94,69,29,44)(22,73,95,70,30,45)(23,74,96,71,31,46)(24,75,89,72,32,47), (1,72,83,24)(2,65,84,17)(3,66,85,18)(4,67,86,19)(5,68,87,20)(6,69,88,21)(7,70,81,22)(8,71,82,23)(9,77,63,26)(10,78,64,27)(11,79,57,28)(12,80,58,29)(13,73,59,30)(14,74,60,31)(15,75,61,32)(16,76,62,25)(33,91,52,41)(34,92,53,42)(35,93,54,43)(36,94,55,44)(37,95,56,45)(38,96,49,46)(39,89,50,47)(40,90,51,48), (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)(10,60)(12,62)(14,64)(16,58)(17,69)(19,71)(21,65)(23,67)(25,80)(27,74)(29,76)(31,78)(34,49)(36,51)(38,53)(40,55)(42,96)(44,90)(46,92)(48,94)>;
G:=Group( (1,39,15,83,50,61)(2,40,16,84,51,62)(3,33,9,85,52,63)(4,34,10,86,53,64)(5,35,11,87,54,57)(6,36,12,88,55,58)(7,37,13,81,56,59)(8,38,14,82,49,60)(17,76,90,65,25,48)(18,77,91,66,26,41)(19,78,92,67,27,42)(20,79,93,68,28,43)(21,80,94,69,29,44)(22,73,95,70,30,45)(23,74,96,71,31,46)(24,75,89,72,32,47), (1,72,83,24)(2,65,84,17)(3,66,85,18)(4,67,86,19)(5,68,87,20)(6,69,88,21)(7,70,81,22)(8,71,82,23)(9,77,63,26)(10,78,64,27)(11,79,57,28)(12,80,58,29)(13,73,59,30)(14,74,60,31)(15,75,61,32)(16,76,62,25)(33,91,52,41)(34,92,53,42)(35,93,54,43)(36,94,55,44)(37,95,56,45)(38,96,49,46)(39,89,50,47)(40,90,51,48), (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)(10,60)(12,62)(14,64)(16,58)(17,69)(19,71)(21,65)(23,67)(25,80)(27,74)(29,76)(31,78)(34,49)(36,51)(38,53)(40,55)(42,96)(44,90)(46,92)(48,94) );
G=PermutationGroup([[(1,39,15,83,50,61),(2,40,16,84,51,62),(3,33,9,85,52,63),(4,34,10,86,53,64),(5,35,11,87,54,57),(6,36,12,88,55,58),(7,37,13,81,56,59),(8,38,14,82,49,60),(17,76,90,65,25,48),(18,77,91,66,26,41),(19,78,92,67,27,42),(20,79,93,68,28,43),(21,80,94,69,29,44),(22,73,95,70,30,45),(23,74,96,71,31,46),(24,75,89,72,32,47)], [(1,72,83,24),(2,65,84,17),(3,66,85,18),(4,67,86,19),(5,68,87,20),(6,69,88,21),(7,70,81,22),(8,71,82,23),(9,77,63,26),(10,78,64,27),(11,79,57,28),(12,80,58,29),(13,73,59,30),(14,74,60,31),(15,75,61,32),(16,76,62,25),(33,91,52,41),(34,92,53,42),(35,93,54,43),(36,94,55,44),(37,95,56,45),(38,96,49,46),(39,89,50,47),(40,90,51,48)], [(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),(10,60),(12,62),(14,64),(16,58),(17,69),(19,71),(21,65),(23,67),(25,80),(27,74),(29,76),(31,78),(34,49),(36,51),(38,53),(40,55),(42,96),(44,90),(46,92),(48,94)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | ··· | 4N | 4O | 4P | 4Q | 4R | 6A | 6B | 6C | 6D | 6E | 8A | ··· | 8H | 8I | ··· | 8P | 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 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 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 | 3 | ··· | 3 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 4 | 4 | 2 | ··· | 2 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | - | |||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D6 | D6 | M4(2) | C4○D4 | C4×S3 | C4×S3 | S3×C8 | D4⋊2S3 | S3×M4(2) |
| kernel | Dic3.5M4(2) | C8×Dic3 | Dic3⋊C8 | C12.55D4 | C3×C22⋊C8 | C2×C4×Dic3 | C4×Dic3 | C22×Dic3 | C2×Dic3 | C22⋊C8 | C2×C8 | C22×C4 | Dic3 | C12 | C2×C4 | C23 | C22 | C4 | C2 |
| # reps | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 4 | 16 | 1 | 2 | 1 | 4 | 4 | 2 | 2 | 8 | 2 | 2 |
Matrix representation of Dic3.5M4(2) ►in GL4(𝔽73) generated by
| 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 22 | 61 | 0 | 0 |
| 10 | 51 | 0 | 0 |
| 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 72 |
| 22 | 0 | 0 | 0 |
| 0 | 22 | 0 | 0 |
| 0 | 0 | 0 | 1 |
| 0 | 0 | 46 | 0 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 72 |
G:=sub<GL(4,GF(73))| [1,1,0,0,72,0,0,0,0,0,1,0,0,0,0,1],[22,10,0,0,61,51,0,0,0,0,72,0,0,0,0,72],[22,0,0,0,0,22,0,0,0,0,0,46,0,0,1,0],[1,0,0,0,0,1,0,0,0,0,1,0,0,0,0,72] >;
Dic3.5M4(2) in GAP, Magma, Sage, TeX
{\rm Dic}_3._5M_4(2) % in TeX
G:=Group("Dic3.5M4(2)"); // GroupNames label
G:=SmallGroup(192,277);
// by ID
G=gap.SmallGroup(192,277);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,219,58,136,6278]);
// 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,b*c=c*b,b*d=d*b,d*c*d=a^3*c^5>;
// generators/relations