direct product, metacyclic, supersoluble, monomial
Aliases: C3×C8⋊Dic3, C24⋊2C12, C24⋊6Dic3, C62.78D4, C12.31Dic6, (C3×C24)⋊8C4, C8⋊2(C3×Dic3), C12.4(C3×Q8), (C2×C24).12C6, (C6×C24).14C2, (C2×C24).34S3, (C2×C6).69D12, C4⋊Dic3.2C6, (C3×C12).22Q8, C4.4(C3×Dic6), C6.2(C3×SD16), C4.6(C6×Dic3), C12.41(C2×C12), C32⋊9(C4.Q8), (C2×C12).432D6, (C3×C6).16SD16, C22.8(C3×D12), C6.14(C24⋊C2), C6.18(C4⋊Dic3), C12.63(C2×Dic3), (C6×C12).316C22, C6.5(C3×C4⋊C4), C3⋊2(C3×C4.Q8), (C2×C8).6(C3×S3), C2.2(C3×C24⋊C2), (C2×C4).71(S3×C6), (C2×C6).17(C3×D4), C2.3(C3×C4⋊Dic3), (C3×C6).31(C4⋊C4), (C3×C12).127(C2×C4), (C2×C12).101(C2×C6), (C3×C4⋊Dic3).21C2, SmallGroup(288,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8⋊Dic3
G = < a,b,c,d | a3=b8=c6=1, d2=c3, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=b3, dcd-1=c-1 >
Subgroups: 186 in 83 conjugacy classes, 50 normal (30 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C6, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C3×C6, C24, C24, C2×Dic3, C2×C12, C2×C12, C4.Q8, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×C24, C3×C24, C6×Dic3, C6×C12, C8⋊Dic3, C3×C4.Q8, C3×C4⋊Dic3, C6×C24, C3×C8⋊Dic3
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, SD16, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C4.Q8, C3×Dic3, S3×C6, C24⋊C2, C4⋊Dic3, C3×C4⋊C4, C3×SD16, C3×Dic6, C3×D12, C6×Dic3, C8⋊Dic3, C3×C4.Q8, C3×C24⋊C2, C3×C4⋊Dic3, C3×C8⋊Dic3
►On 96 points
(1 77 64)(2 78 57)(3 79 58)(4 80 59)(5 73 60)(6 74 61)(7 75 62)(8 76 63)(9 85 43)(10 86 44)(11 87 45)(12 88 46)(13 81 47)(14 82 48)(15 83 41)(16 84 42)(17 67 31)(18 68 32)(19 69 25)(20 70 26)(21 71 27)(22 72 28)(23 65 29)(24 66 30)(33 93 53)(34 94 54)(35 95 55)(36 96 56)(37 89 49)(38 90 50)(39 91 51)(40 92 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 66 64 24 77 30)(2 67 57 17 78 31)(3 68 58 18 79 32)(4 69 59 19 80 25)(5 70 60 20 73 26)(6 71 61 21 74 27)(7 72 62 22 75 28)(8 65 63 23 76 29)(9 90 85 50 43 38)(10 91 86 51 44 39)(11 92 87 52 45 40)(12 93 88 53 46 33)(13 94 81 54 47 34)(14 95 82 55 48 35)(15 96 83 56 41 36)(16 89 84 49 42 37)
(1 50 24 9)(2 53 17 12)(3 56 18 15)(4 51 19 10)(5 54 20 13)(6 49 21 16)(7 52 22 11)(8 55 23 14)(25 44 59 91)(26 47 60 94)(27 42 61 89)(28 45 62 92)(29 48 63 95)(30 43 64 90)(31 46 57 93)(32 41 58 96)(33 67 88 78)(34 70 81 73)(35 65 82 76)(36 68 83 79)(37 71 84 74)(38 66 85 77)(39 69 86 80)(40 72 87 75)
G:=sub<Sym(96)| (1,77,64)(2,78,57)(3,79,58)(4,80,59)(5,73,60)(6,74,61)(7,75,62)(8,76,63)(9,85,43)(10,86,44)(11,87,45)(12,88,46)(13,81,47)(14,82,48)(15,83,41)(16,84,42)(17,67,31)(18,68,32)(19,69,25)(20,70,26)(21,71,27)(22,72,28)(23,65,29)(24,66,30)(33,93,53)(34,94,54)(35,95,55)(36,96,56)(37,89,49)(38,90,50)(39,91,51)(40,92,52), (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), (1,66,64,24,77,30)(2,67,57,17,78,31)(3,68,58,18,79,32)(4,69,59,19,80,25)(5,70,60,20,73,26)(6,71,61,21,74,27)(7,72,62,22,75,28)(8,65,63,23,76,29)(9,90,85,50,43,38)(10,91,86,51,44,39)(11,92,87,52,45,40)(12,93,88,53,46,33)(13,94,81,54,47,34)(14,95,82,55,48,35)(15,96,83,56,41,36)(16,89,84,49,42,37), (1,50,24,9)(2,53,17,12)(3,56,18,15)(4,51,19,10)(5,54,20,13)(6,49,21,16)(7,52,22,11)(8,55,23,14)(25,44,59,91)(26,47,60,94)(27,42,61,89)(28,45,62,92)(29,48,63,95)(30,43,64,90)(31,46,57,93)(32,41,58,96)(33,67,88,78)(34,70,81,73)(35,65,82,76)(36,68,83,79)(37,71,84,74)(38,66,85,77)(39,69,86,80)(40,72,87,75)>;
G:=Group( (1,77,64)(2,78,57)(3,79,58)(4,80,59)(5,73,60)(6,74,61)(7,75,62)(8,76,63)(9,85,43)(10,86,44)(11,87,45)(12,88,46)(13,81,47)(14,82,48)(15,83,41)(16,84,42)(17,67,31)(18,68,32)(19,69,25)(20,70,26)(21,71,27)(22,72,28)(23,65,29)(24,66,30)(33,93,53)(34,94,54)(35,95,55)(36,96,56)(37,89,49)(38,90,50)(39,91,51)(40,92,52), (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), (1,66,64,24,77,30)(2,67,57,17,78,31)(3,68,58,18,79,32)(4,69,59,19,80,25)(5,70,60,20,73,26)(6,71,61,21,74,27)(7,72,62,22,75,28)(8,65,63,23,76,29)(9,90,85,50,43,38)(10,91,86,51,44,39)(11,92,87,52,45,40)(12,93,88,53,46,33)(13,94,81,54,47,34)(14,95,82,55,48,35)(15,96,83,56,41,36)(16,89,84,49,42,37), (1,50,24,9)(2,53,17,12)(3,56,18,15)(4,51,19,10)(5,54,20,13)(6,49,21,16)(7,52,22,11)(8,55,23,14)(25,44,59,91)(26,47,60,94)(27,42,61,89)(28,45,62,92)(29,48,63,95)(30,43,64,90)(31,46,57,93)(32,41,58,96)(33,67,88,78)(34,70,81,73)(35,65,82,76)(36,68,83,79)(37,71,84,74)(38,66,85,77)(39,69,86,80)(40,72,87,75) );
G=PermutationGroup([[(1,77,64),(2,78,57),(3,79,58),(4,80,59),(5,73,60),(6,74,61),(7,75,62),(8,76,63),(9,85,43),(10,86,44),(11,87,45),(12,88,46),(13,81,47),(14,82,48),(15,83,41),(16,84,42),(17,67,31),(18,68,32),(19,69,25),(20,70,26),(21,71,27),(22,72,28),(23,65,29),(24,66,30),(33,93,53),(34,94,54),(35,95,55),(36,96,56),(37,89,49),(38,90,50),(39,91,51),(40,92,52)], [(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)], [(1,66,64,24,77,30),(2,67,57,17,78,31),(3,68,58,18,79,32),(4,69,59,19,80,25),(5,70,60,20,73,26),(6,71,61,21,74,27),(7,72,62,22,75,28),(8,65,63,23,76,29),(9,90,85,50,43,38),(10,91,86,51,44,39),(11,92,87,52,45,40),(12,93,88,53,46,33),(13,94,81,54,47,34),(14,95,82,55,48,35),(15,96,83,56,41,36),(16,89,84,49,42,37)], [(1,50,24,9),(2,53,17,12),(3,56,18,15),(4,51,19,10),(5,54,20,13),(6,49,21,16),(7,52,22,11),(8,55,23,14),(25,44,59,91),(26,47,60,94),(27,42,61,89),(28,45,62,92),(29,48,63,95),(30,43,64,90),(31,46,57,93),(32,41,58,96),(33,67,88,78),(34,70,81,73),(35,65,82,76),(36,68,83,79),(37,71,84,74),(38,66,85,77),(39,69,86,80),(40,72,87,75)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | ||||||||||||||||
| image | C1 | C2 | C2 | C3 | C4 | C6 | C6 | C12 | S3 | Q8 | D4 | Dic3 | D6 | SD16 | C3×S3 | Dic6 | C3×Q8 | D12 | C3×D4 | C3×Dic3 | S3×C6 | C24⋊C2 | C3×SD16 | C3×Dic6 | C3×D12 | C3×C24⋊C2 |
| kernel | C3×C8⋊Dic3 | C3×C4⋊Dic3 | C6×C24 | C8⋊Dic3 | C3×C24 | C4⋊Dic3 | C2×C24 | C24 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C8 | C2×C4 | C6 | C6 | C4 | C22 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 8 | 8 | 4 | 4 | 16 |
Matrix representation of C3×C8⋊Dic3 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 8 |
| 72 | 0 | 0 |
| 0 | 10 | 0 |
| 0 | 0 | 51 |
| 72 | 0 | 0 |
| 0 | 8 | 0 |
| 0 | 0 | 64 |
| 46 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(73))| [8,0,0,0,8,0,0,0,8],[72,0,0,0,10,0,0,0,51],[72,0,0,0,8,0,0,0,64],[46,0,0,0,0,1,0,1,0] >;
C3×C8⋊Dic3 in GAP, Magma, Sage, TeX
C_3\times C_8\rtimes {\rm Dic}_3 % in TeX
G:=Group("C3xC8:Dic3"); // GroupNames label
G:=SmallGroup(288,251);
// by ID
G=gap.SmallGroup(288,251);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,176,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=c^6=1,d^2=c^3,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b^3,d*c*d^-1=c^-1>;
// generators/relations