metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: D4⋊3Dic6, C12.50D8, C42.44D6, (C3×D4)⋊3Q8, (C4×D4).2S3, C6.50(C2×D8), C4⋊C4.238D6, C12⋊C8⋊17C2, (D4×C12).2C2, C3⋊4(D4⋊Q8), (C2×C12).57D4, C12.25(C2×Q8), C4.9(C2×Dic6), C4.23(D4⋊S3), C12⋊2Q8⋊14C2, (C2×D4).185D6, C6.Q16⋊30C2, C4.59(C4○D12), C12.45(C4○D4), (C4×C12).78C22, D4⋊Dic3.7C2, C6.61(C22⋊Q8), (C2×C12).332C23, C2.7(Q8.14D6), (C6×D4).227C22, C6.107(C8.C22), C4⋊Dic3.136C22, C2.12(C12.48D4), C2.6(C2×D4⋊S3), (C2×C6).463(C2×D4), (C2×C3⋊C8).89C22, (C2×C4).243(C3⋊D4), (C3×C4⋊C4).269C22, (C2×C4).432(C22×S3), C22.146(C2×C3⋊D4), SmallGroup(192,566)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.50D8
G = < a,b,c | a12=b8=1, c2=a6, bab-1=cac-1=a-1, cbc-1=a6b-1 >
Subgroups: 280 in 108 conjugacy classes, 47 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, C6, C6, C8, C2×C4, C2×C4, D4, D4, Q8, C23, Dic3, C12, C12, C12, C2×C6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C22×C4, C2×D4, C2×Q8, C3⋊C8, Dic6, C2×Dic3, C2×C12, C2×C12, C3×D4, C3×D4, C22×C6, D4⋊C4, C4⋊C8, C2.D8, C4×D4, C4⋊Q8, C2×C3⋊C8, C4⋊Dic3, C4⋊Dic3, C4×C12, C3×C22⋊C4, C3×C4⋊C4, C2×Dic6, C22×C12, C6×D4, D4⋊Q8, C12⋊C8, C6.Q16, D4⋊Dic3, C12⋊2Q8, D4×C12, C12.50D8
Quotients: C1, C2, C22, S3, D4, Q8, C23, D6, D8, C2×D4, C2×Q8, C4○D4, Dic6, C3⋊D4, C22×S3, C22⋊Q8, C2×D8, C8.C22, D4⋊S3, C2×Dic6, C4○D12, C2×C3⋊D4, D4⋊Q8, C12.48D4, C2×D4⋊S3, Q8.14D6, C12.50D8
►On 96 points
(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 16 32 42 66 51 88 79)(2 15 33 41 67 50 89 78)(3 14 34 40 68 49 90 77)(4 13 35 39 69 60 91 76)(5 24 36 38 70 59 92 75)(6 23 25 37 71 58 93 74)(7 22 26 48 72 57 94 73)(8 21 27 47 61 56 95 84)(9 20 28 46 62 55 96 83)(10 19 29 45 63 54 85 82)(11 18 30 44 64 53 86 81)(12 17 31 43 65 52 87 80)
(1 73 7 79)(2 84 8 78)(3 83 9 77)(4 82 10 76)(5 81 11 75)(6 80 12 74)(13 91 19 85)(14 90 20 96)(15 89 21 95)(16 88 22 94)(17 87 23 93)(18 86 24 92)(25 52 31 58)(26 51 32 57)(27 50 33 56)(28 49 34 55)(29 60 35 54)(30 59 36 53)(37 71 43 65)(38 70 44 64)(39 69 45 63)(40 68 46 62)(41 67 47 61)(42 66 48 72)
G:=sub<Sym(96)| (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), (1,16,32,42,66,51,88,79)(2,15,33,41,67,50,89,78)(3,14,34,40,68,49,90,77)(4,13,35,39,69,60,91,76)(5,24,36,38,70,59,92,75)(6,23,25,37,71,58,93,74)(7,22,26,48,72,57,94,73)(8,21,27,47,61,56,95,84)(9,20,28,46,62,55,96,83)(10,19,29,45,63,54,85,82)(11,18,30,44,64,53,86,81)(12,17,31,43,65,52,87,80), (1,73,7,79)(2,84,8,78)(3,83,9,77)(4,82,10,76)(5,81,11,75)(6,80,12,74)(13,91,19,85)(14,90,20,96)(15,89,21,95)(16,88,22,94)(17,87,23,93)(18,86,24,92)(25,52,31,58)(26,51,32,57)(27,50,33,56)(28,49,34,55)(29,60,35,54)(30,59,36,53)(37,71,43,65)(38,70,44,64)(39,69,45,63)(40,68,46,62)(41,67,47,61)(42,66,48,72)>;
G:=Group( (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), (1,16,32,42,66,51,88,79)(2,15,33,41,67,50,89,78)(3,14,34,40,68,49,90,77)(4,13,35,39,69,60,91,76)(5,24,36,38,70,59,92,75)(6,23,25,37,71,58,93,74)(7,22,26,48,72,57,94,73)(8,21,27,47,61,56,95,84)(9,20,28,46,62,55,96,83)(10,19,29,45,63,54,85,82)(11,18,30,44,64,53,86,81)(12,17,31,43,65,52,87,80), (1,73,7,79)(2,84,8,78)(3,83,9,77)(4,82,10,76)(5,81,11,75)(6,80,12,74)(13,91,19,85)(14,90,20,96)(15,89,21,95)(16,88,22,94)(17,87,23,93)(18,86,24,92)(25,52,31,58)(26,51,32,57)(27,50,33,56)(28,49,34,55)(29,60,35,54)(30,59,36,53)(37,71,43,65)(38,70,44,64)(39,69,45,63)(40,68,46,62)(41,67,47,61)(42,66,48,72) );
G=PermutationGroup([[(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)], [(1,16,32,42,66,51,88,79),(2,15,33,41,67,50,89,78),(3,14,34,40,68,49,90,77),(4,13,35,39,69,60,91,76),(5,24,36,38,70,59,92,75),(6,23,25,37,71,58,93,74),(7,22,26,48,72,57,94,73),(8,21,27,47,61,56,95,84),(9,20,28,46,62,55,96,83),(10,19,29,45,63,54,85,82),(11,18,30,44,64,53,86,81),(12,17,31,43,65,52,87,80)], [(1,73,7,79),(2,84,8,78),(3,83,9,77),(4,82,10,76),(5,81,11,75),(6,80,12,74),(13,91,19,85),(14,90,20,96),(15,89,21,95),(16,88,22,94),(17,87,23,93),(18,86,24,92),(25,52,31,58),(26,51,32,57),(27,50,33,56),(28,49,34,55),(29,60,35,54),(30,59,36,53),(37,71,43,65),(38,70,44,64),(39,69,45,63),(40,68,46,62),(41,67,47,61),(42,66,48,72)]])
39 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
39 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | - | + | + | + | + | - | - | + | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | Q8 | D6 | D6 | D6 | D8 | C4○D4 | C3⋊D4 | Dic6 | C4○D12 | C8.C22 | D4⋊S3 | Q8.14D6 |
| kernel | C12.50D8 | C12⋊C8 | C6.Q16 | D4⋊Dic3 | C12⋊2Q8 | D4×C12 | C4×D4 | C2×C12 | C3×D4 | C42 | C4⋊C4 | C2×D4 | C12 | C12 | C2×C4 | D4 | C4 | C6 | C4 | C2 |
| # reps | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 1 | 4 | 2 | 4 | 4 | 4 | 1 | 2 | 2 |
Matrix representation of C12.50D8 ►in GL4(𝔽73) generated by
| 72 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 3 | 0 |
| 0 | 0 | 33 | 49 |
| 16 | 57 | 0 | 0 |
| 16 | 16 | 0 | 0 |
| 0 | 0 | 32 | 7 |
| 0 | 0 | 52 | 41 |
| 16 | 57 | 0 | 0 |
| 57 | 57 | 0 | 0 |
| 0 | 0 | 32 | 7 |
| 0 | 0 | 10 | 41 |
G:=sub<GL(4,GF(73))| [72,0,0,0,0,72,0,0,0,0,3,33,0,0,0,49],[16,16,0,0,57,16,0,0,0,0,32,52,0,0,7,41],[16,57,0,0,57,57,0,0,0,0,32,10,0,0,7,41] >;
C12.50D8 in GAP, Magma, Sage, TeX
C_{12}._{50}D_8 % in TeX
G:=Group("C12.50D8"); // GroupNames label
G:=SmallGroup(192,566);
// by ID
G=gap.SmallGroup(192,566);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,253,120,254,1123,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^6,b*a*b^-1=c*a*c^-1=a^-1,c*b*c^-1=a^6*b^-1>;
// generators/relations