direct product, metabelian, supersoluble, monomial
Aliases: C3×C8.6D6, D24.2C6, C24.54D6, C32⋊9SD32, C3⋊C16⋊3C6, C8.6(S3×C6), C24.4(C2×C6), (C3×Q16)⋊1C6, (C3×Q16)⋊5S3, Q16⋊1(C3×S3), C3⋊3(C3×SD32), C6.10(C3×D8), C12.5(C3×D4), (C3×C6).32D8, (C3×D24).4C2, (C3×C12).43D4, C6.32(D4⋊S3), (C32×Q16)⋊1C2, C12.85(C3⋊D4), (C3×C24).15C22, (C3×C3⋊C16)⋊6C2, C2.6(C3×D4⋊S3), C4.3(C3×C3⋊D4), SmallGroup(288,262)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C8.6D6
G = < a,b,c,d | a3=b8=1, c6=b4, d2=b3, ab=ba, ac=ca, ad=da, cbc-1=b-1, bd=db, dcd-1=b-1c5 >
Subgroups: 202 in 61 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, C12, C12, D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24, C24, D12, C3×D4, C3×Q8, SD32, C3×C12, C3×C12, S3×C6, C3⋊C16, C48, D24, C3×D8, C3×Q16, C3×Q16, C3×C24, C3×D12, Q8×C32, C8.6D6, C3×SD32, C3×C3⋊C16, C3×D24, C32×Q16, C3×C8.6D6
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, C3⋊D4, C3×D4, SD32, S3×C6, D4⋊S3, C3×D8, C3×C3⋊D4, C8.6D6, C3×SD32, C3×D4⋊S3, C3×C8.6D6
►On 96 points
Generators in S96
(1 39 75)(2 40 76)(3 41 77)(4 42 78)(5 43 79)(6 44 80)(7 45 65)(8 46 66)(9 47 67)(10 48 68)(11 33 69)(12 34 70)(13 35 71)(14 36 72)(15 37 73)(16 38 74)(17 58 83)(18 59 84)(19 60 85)(20 61 86)(21 62 87)(22 63 88)(23 64 89)(24 49 90)(25 50 91)(26 51 92)(27 52 93)(28 53 94)(29 54 95)(30 55 96)(31 56 81)(32 57 82)
(1 7 13 3 9 15 5 11)(2 8 14 4 10 16 6 12)(17 23 29 19 25 31 21 27)(18 24 30 20 26 32 22 28)(33 39 45 35 41 47 37 43)(34 40 46 36 42 48 38 44)(49 55 61 51 57 63 53 59)(50 56 62 52 58 64 54 60)(65 71 77 67 73 79 69 75)(66 72 78 68 74 80 70 76)(81 87 93 83 89 95 85 91)(82 88 94 84 90 96 86 92)
(1 49 47 82 75 24 9 57 39 90 67 32)(2 31 68 89 40 56 10 23 76 81 48 64)(3 63 33 96 77 22 11 55 41 88 69 30)(4 29 70 87 42 54 12 21 78 95 34 62)(5 61 35 94 79 20 13 53 43 86 71 28)(6 27 72 85 44 52 14 19 80 93 36 60)(7 59 37 92 65 18 15 51 45 84 73 26)(8 25 74 83 46 50 16 17 66 91 38 58)
(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)
G:=sub<Sym(96)| (1,39,75)(2,40,76)(3,41,77)(4,42,78)(5,43,79)(6,44,80)(7,45,65)(8,46,66)(9,47,67)(10,48,68)(11,33,69)(12,34,70)(13,35,71)(14,36,72)(15,37,73)(16,38,74)(17,58,83)(18,59,84)(19,60,85)(20,61,86)(21,62,87)(22,63,88)(23,64,89)(24,49,90)(25,50,91)(26,51,92)(27,52,93)(28,53,94)(29,54,95)(30,55,96)(31,56,81)(32,57,82), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92), (1,49,47,82,75,24,9,57,39,90,67,32)(2,31,68,89,40,56,10,23,76,81,48,64)(3,63,33,96,77,22,11,55,41,88,69,30)(4,29,70,87,42,54,12,21,78,95,34,62)(5,61,35,94,79,20,13,53,43,86,71,28)(6,27,72,85,44,52,14,19,80,93,36,60)(7,59,37,92,65,18,15,51,45,84,73,26)(8,25,74,83,46,50,16,17,66,91,38,58), (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)>;
G:=Group( (1,39,75)(2,40,76)(3,41,77)(4,42,78)(5,43,79)(6,44,80)(7,45,65)(8,46,66)(9,47,67)(10,48,68)(11,33,69)(12,34,70)(13,35,71)(14,36,72)(15,37,73)(16,38,74)(17,58,83)(18,59,84)(19,60,85)(20,61,86)(21,62,87)(22,63,88)(23,64,89)(24,49,90)(25,50,91)(26,51,92)(27,52,93)(28,53,94)(29,54,95)(30,55,96)(31,56,81)(32,57,82), (1,7,13,3,9,15,5,11)(2,8,14,4,10,16,6,12)(17,23,29,19,25,31,21,27)(18,24,30,20,26,32,22,28)(33,39,45,35,41,47,37,43)(34,40,46,36,42,48,38,44)(49,55,61,51,57,63,53,59)(50,56,62,52,58,64,54,60)(65,71,77,67,73,79,69,75)(66,72,78,68,74,80,70,76)(81,87,93,83,89,95,85,91)(82,88,94,84,90,96,86,92), (1,49,47,82,75,24,9,57,39,90,67,32)(2,31,68,89,40,56,10,23,76,81,48,64)(3,63,33,96,77,22,11,55,41,88,69,30)(4,29,70,87,42,54,12,21,78,95,34,62)(5,61,35,94,79,20,13,53,43,86,71,28)(6,27,72,85,44,52,14,19,80,93,36,60)(7,59,37,92,65,18,15,51,45,84,73,26)(8,25,74,83,46,50,16,17,66,91,38,58), (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) );
G=PermutationGroup([[(1,39,75),(2,40,76),(3,41,77),(4,42,78),(5,43,79),(6,44,80),(7,45,65),(8,46,66),(9,47,67),(10,48,68),(11,33,69),(12,34,70),(13,35,71),(14,36,72),(15,37,73),(16,38,74),(17,58,83),(18,59,84),(19,60,85),(20,61,86),(21,62,87),(22,63,88),(23,64,89),(24,49,90),(25,50,91),(26,51,92),(27,52,93),(28,53,94),(29,54,95),(30,55,96),(31,56,81),(32,57,82)], [(1,7,13,3,9,15,5,11),(2,8,14,4,10,16,6,12),(17,23,29,19,25,31,21,27),(18,24,30,20,26,32,22,28),(33,39,45,35,41,47,37,43),(34,40,46,36,42,48,38,44),(49,55,61,51,57,63,53,59),(50,56,62,52,58,64,54,60),(65,71,77,67,73,79,69,75),(66,72,78,68,74,80,70,76),(81,87,93,83,89,95,85,91),(82,88,94,84,90,96,86,92)], [(1,49,47,82,75,24,9,57,39,90,67,32),(2,31,68,89,40,56,10,23,76,81,48,64),(3,63,33,96,77,22,11,55,41,88,69,30),(4,29,70,87,42,54,12,21,78,95,34,62),(5,61,35,94,79,20,13,53,43,86,71,28),(6,27,72,85,44,52,14,19,80,93,36,60),(7,59,37,92,65,18,15,51,45,84,73,26),(8,25,74,83,46,50,16,17,66,91,38,58)], [(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)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | 12B | 12C | 12D | 12E | 12F | ··· | 12M | 16A | 16B | 16C | 16D | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 16 | 16 | 16 | 16 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 24 | 1 | 1 | 2 | 2 | 2 | 2 | 8 | 1 | 1 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 8 | ··· | 8 | 6 | 6 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | ··· | 6 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | C3⋊D4 | C3×D4 | SD32 | S3×C6 | C3×D8 | C3×C3⋊D4 | C3×SD32 | D4⋊S3 | C8.6D6 | C3×D4⋊S3 | C3×C8.6D6 |
| kernel | C3×C8.6D6 | C3×C3⋊C16 | C3×D24 | C32×Q16 | C8.6D6 | C3⋊C16 | D24 | C3×Q16 | C3×Q16 | C3×C12 | C24 | C3×C6 | Q16 | C12 | C12 | C32 | C8 | C6 | C4 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×C8.6D6 ►in GL4(𝔽7) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 4 | 0 |
| 0 | 0 | 0 | 4 |
| 6 | 3 | 0 | 2 |
| 1 | 6 | 1 | 1 |
| 2 | 2 | 2 | 3 |
| 3 | 4 | 2 | 6 |
| 2 | 6 | 5 | 4 |
| 5 | 2 | 3 | 5 |
| 6 | 2 | 6 | 3 |
| 1 | 3 | 4 | 4 |
| 2 | 0 | 6 | 6 |
| 5 | 6 | 0 | 6 |
| 4 | 6 | 4 | 1 |
| 4 | 5 | 1 | 2 |
G:=sub<GL(4,GF(7))| [4,0,0,0,0,4,0,0,0,0,4,0,0,0,0,4],[6,1,2,3,3,6,2,4,0,1,2,2,2,1,3,6],[2,5,6,1,6,2,2,3,5,3,6,4,4,5,3,4],[2,5,4,4,0,6,6,5,6,0,4,1,6,6,1,2] >;
C3×C8.6D6 in GAP, Magma, Sage, TeX
C_3\times C_8._6D_6
% in TeX
G:=Group("C3xC8.6D6"); // GroupNames label
G:=SmallGroup(288,262);
// by ID
G=gap.SmallGroup(288,262);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,344,1011,514,192,2524,1271,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^8=1,c^6=b^4,d^2=b^3,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c^-1=b^-1,b*d=d*b,d*c*d^-1=b^-1*c^5>;
// generators/relations