metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D12.2C8, C8.26D12, C24.51D4, M5(2)⋊4S3, Dic6.2C8, C4.3(S3×C8), C12.6(C2×C8), C3⋊2(D4.C8), C4○D12.2C4, C8○D12.4C2, (C2×C8).267D6, C2.11(D6⋊C8), C4.43(D6⋊C4), C8.51(C3⋊D4), (C3×M5(2))⋊8C2, (C2×C6).5M4(2), C4.Dic3.3C4, C6.10(C22⋊C8), C12.58(C22⋊C4), (C2×C24).264C22, C22.1(C8⋊S3), (C2×C3⋊C16)⋊13C2, (C2×C4).67(C4×S3), (C2×C12).53(C2×C4), SmallGroup(192,74)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for Dic6.C8
G = < a,b,c | a12=1, b2=c8=a6, bab-1=a-1, cac-1=a7, cbc-1=a9b >
Smallest permutation representation of Dic6.C8
►On 96 points
Generators in S96
(1 33 56 96 22 73 9 41 64 88 30 65)(2 42 57 89 23 66 10 34 49 81 31 74)(3 35 58 82 24 75 11 43 50 90 32 67)(4 44 59 91 25 68 12 36 51 83 17 76)(5 37 60 84 26 77 13 45 52 92 18 69)(6 46 61 93 27 70 14 38 53 85 19 78)(7 39 62 86 28 79 15 47 54 94 20 71)(8 48 63 95 29 72 16 40 55 87 21 80)
(1 5 9 13)(2 85 10 93)(3 7 11 15)(4 87 12 95)(6 89 14 81)(8 91 16 83)(17 80 25 72)(18 64 26 56)(19 66 27 74)(20 50 28 58)(21 68 29 76)(22 52 30 60)(23 70 31 78)(24 54 32 62)(33 69 41 77)(34 61 42 53)(35 71 43 79)(36 63 44 55)(37 73 45 65)(38 49 46 57)(39 75 47 67)(40 51 48 59)(82 94 90 86)(84 96 92 88)
(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)| (1,33,56,96,22,73,9,41,64,88,30,65)(2,42,57,89,23,66,10,34,49,81,31,74)(3,35,58,82,24,75,11,43,50,90,32,67)(4,44,59,91,25,68,12,36,51,83,17,76)(5,37,60,84,26,77,13,45,52,92,18,69)(6,46,61,93,27,70,14,38,53,85,19,78)(7,39,62,86,28,79,15,47,54,94,20,71)(8,48,63,95,29,72,16,40,55,87,21,80), (1,5,9,13)(2,85,10,93)(3,7,11,15)(4,87,12,95)(6,89,14,81)(8,91,16,83)(17,80,25,72)(18,64,26,56)(19,66,27,74)(20,50,28,58)(21,68,29,76)(22,52,30,60)(23,70,31,78)(24,54,32,62)(33,69,41,77)(34,61,42,53)(35,71,43,79)(36,63,44,55)(37,73,45,65)(38,49,46,57)(39,75,47,67)(40,51,48,59)(82,94,90,86)(84,96,92,88), (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( (1,33,56,96,22,73,9,41,64,88,30,65)(2,42,57,89,23,66,10,34,49,81,31,74)(3,35,58,82,24,75,11,43,50,90,32,67)(4,44,59,91,25,68,12,36,51,83,17,76)(5,37,60,84,26,77,13,45,52,92,18,69)(6,46,61,93,27,70,14,38,53,85,19,78)(7,39,62,86,28,79,15,47,54,94,20,71)(8,48,63,95,29,72,16,40,55,87,21,80), (1,5,9,13)(2,85,10,93)(3,7,11,15)(4,87,12,95)(6,89,14,81)(8,91,16,83)(17,80,25,72)(18,64,26,56)(19,66,27,74)(20,50,28,58)(21,68,29,76)(22,52,30,60)(23,70,31,78)(24,54,32,62)(33,69,41,77)(34,61,42,53)(35,71,43,79)(36,63,44,55)(37,73,45,65)(38,49,46,57)(39,75,47,67)(40,51,48,59)(82,94,90,86)(84,96,92,88), (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([[(1,33,56,96,22,73,9,41,64,88,30,65),(2,42,57,89,23,66,10,34,49,81,31,74),(3,35,58,82,24,75,11,43,50,90,32,67),(4,44,59,91,25,68,12,36,51,83,17,76),(5,37,60,84,26,77,13,45,52,92,18,69),(6,46,61,93,27,70,14,38,53,85,19,78),(7,39,62,86,28,79,15,47,54,94,20,71),(8,48,63,95,29,72,16,40,55,87,21,80)], [(1,5,9,13),(2,85,10,93),(3,7,11,15),(4,87,12,95),(6,89,14,81),(8,91,16,83),(17,80,25,72),(18,64,26,56),(19,66,27,74),(20,50,28,58),(21,68,29,76),(22,52,30,60),(23,70,31,78),(24,54,32,62),(33,69,41,77),(34,61,42,53),(35,71,43,79),(36,63,44,55),(37,73,45,65),(38,49,46,57),(39,75,47,67),(40,51,48,59),(82,94,90,86),(84,96,92,88)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 6A | 6B | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 16A | 16B | 16C | 16D | 16E | ··· | 16L | 24A | 24B | 24C | 24D | 24E | 24F | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | ··· | 16 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 2 | 12 | 2 | 1 | 1 | 2 | 12 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 12 | 12 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 |
| type | + | + | + | + | + | + | + | + | |||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | D4 | D6 | M4(2) | D12 | C3⋊D4 | C4×S3 | S3×C8 | C8⋊S3 | D4.C8 | Dic6.C8 |
| kernel | Dic6.C8 | C2×C3⋊C16 | C3×M5(2) | C8○D12 | C4.Dic3 | C4○D12 | Dic6 | D12 | M5(2) | C24 | C2×C8 | C2×C6 | C8 | C8 | C2×C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 8 | 4 |
Matrix representation of Dic6.C8 ►in GL4(𝔽97) generated by
| 96 | 1 | 0 | 0 |
| 96 | 0 | 0 | 0 |
| 0 | 0 | 96 | 3 |
| 0 | 0 | 64 | 1 |
| 96 | 1 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 22 | 0 |
| 0 | 0 | 47 | 75 |
| 64 | 0 | 0 | 0 |
| 0 | 64 | 0 | 0 |
| 0 | 0 | 0 | 26 |
| 0 | 0 | 92 | 0 |
G:=sub<GL(4,GF(97))| [96,96,0,0,1,0,0,0,0,0,96,64,0,0,3,1],[96,0,0,0,1,1,0,0,0,0,22,47,0,0,0,75],[64,0,0,0,0,64,0,0,0,0,0,92,0,0,26,0] >;
Dic6.C8 in GAP, Magma, Sage, TeX
{\rm Dic}_6.C_8 % in TeX
G:=Group("Dic6.C8"); // GroupNames label
G:=SmallGroup(192,74);
// by ID
G=gap.SmallGroup(192,74);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,141,36,100,1123,570,136,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=1,b^2=c^8=a^6,b*a*b^-1=a^-1,c*a*c^-1=a^7,c*b*c^-1=a^9*b>;
// generators/relations
Export