metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8.2D12, C24.30D4, C4.Q8⋊4S3, C4⋊C4.41D6, (C2×C8).62D6, C4.52(C2×D12), C3⋊2(C8.D4), C12.132(C2×D4), C4.D12.5C2, (C2×Dic12)⋊24C2, C12.32(C4○D4), C6.SD16⋊17C2, C6.45(C4⋊D4), C4.5(Q8⋊3S3), (C2×Dic3).43D4, (C22×S3).26D4, C22.219(S3×D4), C2.18(C12⋊D4), (C2×C12).283C23, (C2×C24).111C22, C2.24(D4.D6), C6.43(C8.C22), (C2×Dic6).84C22, (C3×C4.Q8)⋊4C2, (C2×C8⋊S3).3C2, (C2×C6).288(C2×D4), (C2×C3⋊C8).60C22, (S3×C2×C4).35C22, (C3×C4⋊C4).76C22, (C2×C4).386(C22×S3), SmallGroup(192,426)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C8.2D12
G = < a,b,c | a8=b12=1, c2=a4, bab-1=a3, cac-1=a-1, cbc-1=a4b-1 >
Subgroups: 320 in 110 conjugacy classes, 41 normal (23 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C3⋊C8, C24, Dic6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, C8⋊S3, Dic12, C2×C3⋊C8, C4⋊Dic3, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C8.D4, C6.SD16, C3×C4.Q8, C4.D12, C2×C8⋊S3, C2×Dic12, C8.2D12
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C22×S3, C4⋊D4, C8.C22, C2×D12, S3×D4, Q8⋊3S3, C8.D4, C12⋊D4, D4.D6, C8.2D12
Character table of C8.2D12
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D | |
| size | 1 | 1 | 1 | 1 | 12 | 2 | 2 | 2 | 8 | 8 | 12 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ7 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | linear of order 2 |
| ρ9 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | -2 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from D4 |
| ρ11 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from S3 |
| ρ12 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | 2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ13 | 2 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | -2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ14 | 2 | 2 | 2 | 2 | -2 | 2 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 2 | 2 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ15 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 0 | 0 | -1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | orthogonal lifted from D6 |
| ρ16 | 2 | 2 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | -2 | 0 | 0 | 0 | -1 | -1 | -1 | -2 | -2 | 0 | 0 | -1 | -1 | 1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | orthogonal lifted from D6 |
| ρ17 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | -√3 | √3 | √3 | -√3 | -1 | 1 | 1 | -1 | orthogonal lifted from D12 |
| ρ18 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | 2 | -2 | 0 | 0 | -1 | 1 | √3 | -√3 | -√3 | √3 | -1 | 1 | 1 | -1 | orthogonal lifted from D12 |
| ρ19 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | √3 | -√3 | √3 | -√3 | 1 | -1 | -1 | 1 | orthogonal lifted from D12 |
| ρ20 | 2 | -2 | -2 | 2 | 0 | -1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | -1 | -2 | 2 | 0 | 0 | -1 | 1 | -√3 | √3 | -√3 | √3 | 1 | -1 | -1 | 1 | orthogonal lifted from D12 |
| ρ21 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | 2i | -2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ22 | 2 | -2 | -2 | 2 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | 2 | 0 | 0 | -2i | 2i | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from C4○D4 |
| ρ23 | 4 | 4 | 4 | 4 | 0 | -2 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | -2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from S3×D4 |
| ρ24 | 4 | -4 | -4 | 4 | 0 | -2 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from Q8⋊3S3, Schur index 2 |
| ρ25 | 4 | -4 | 4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4 | -4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ26 | 4 | 4 | -4 | -4 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -4 | 4 | -4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from C8.C22, Schur index 2 |
| ρ27 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | √6 | -√6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ28 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | √6 | -√6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ29 | 4 | -4 | 4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | 2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -√6 | -√6 | √6 | √6 | symplectic lifted from D4.D6, Schur index 2 |
| ρ30 | 4 | 4 | -4 | -4 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | √6 | -√6 | √6 | -√6 | symplectic lifted from D4.D6, Schur index 2 |
►On 96 points
Generators in S96
(1 40 65 16 57 92 31 73)(2 17 32 41 58 74 66 93)(3 42 67 18 59 94 33 75)(4 19 34 43 60 76 68 95)(5 44 69 20 49 96 35 77)(6 21 36 45 50 78 70 85)(7 46 71 22 51 86 25 79)(8 23 26 47 52 80 72 87)(9 48 61 24 53 88 27 81)(10 13 28 37 54 82 62 89)(11 38 63 14 55 90 29 83)(12 15 30 39 56 84 64 91)
(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 56 57 12)(2 11 58 55)(3 54 59 10)(4 9 60 53)(5 52 49 8)(6 7 50 51)(13 75 82 18)(14 17 83 74)(15 73 84 16)(19 81 76 24)(20 23 77 80)(21 79 78 22)(25 70 71 36)(26 35 72 69)(27 68 61 34)(28 33 62 67)(29 66 63 32)(30 31 64 65)(37 94 89 42)(38 41 90 93)(39 92 91 40)(43 88 95 48)(44 47 96 87)(45 86 85 46)
G:=sub<Sym(96)| (1,40,65,16,57,92,31,73)(2,17,32,41,58,74,66,93)(3,42,67,18,59,94,33,75)(4,19,34,43,60,76,68,95)(5,44,69,20,49,96,35,77)(6,21,36,45,50,78,70,85)(7,46,71,22,51,86,25,79)(8,23,26,47,52,80,72,87)(9,48,61,24,53,88,27,81)(10,13,28,37,54,82,62,89)(11,38,63,14,55,90,29,83)(12,15,30,39,56,84,64,91), (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), (1,56,57,12)(2,11,58,55)(3,54,59,10)(4,9,60,53)(5,52,49,8)(6,7,50,51)(13,75,82,18)(14,17,83,74)(15,73,84,16)(19,81,76,24)(20,23,77,80)(21,79,78,22)(25,70,71,36)(26,35,72,69)(27,68,61,34)(28,33,62,67)(29,66,63,32)(30,31,64,65)(37,94,89,42)(38,41,90,93)(39,92,91,40)(43,88,95,48)(44,47,96,87)(45,86,85,46)>;
G:=Group( (1,40,65,16,57,92,31,73)(2,17,32,41,58,74,66,93)(3,42,67,18,59,94,33,75)(4,19,34,43,60,76,68,95)(5,44,69,20,49,96,35,77)(6,21,36,45,50,78,70,85)(7,46,71,22,51,86,25,79)(8,23,26,47,52,80,72,87)(9,48,61,24,53,88,27,81)(10,13,28,37,54,82,62,89)(11,38,63,14,55,90,29,83)(12,15,30,39,56,84,64,91), (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), (1,56,57,12)(2,11,58,55)(3,54,59,10)(4,9,60,53)(5,52,49,8)(6,7,50,51)(13,75,82,18)(14,17,83,74)(15,73,84,16)(19,81,76,24)(20,23,77,80)(21,79,78,22)(25,70,71,36)(26,35,72,69)(27,68,61,34)(28,33,62,67)(29,66,63,32)(30,31,64,65)(37,94,89,42)(38,41,90,93)(39,92,91,40)(43,88,95,48)(44,47,96,87)(45,86,85,46) );
G=PermutationGroup([[(1,40,65,16,57,92,31,73),(2,17,32,41,58,74,66,93),(3,42,67,18,59,94,33,75),(4,19,34,43,60,76,68,95),(5,44,69,20,49,96,35,77),(6,21,36,45,50,78,70,85),(7,46,71,22,51,86,25,79),(8,23,26,47,52,80,72,87),(9,48,61,24,53,88,27,81),(10,13,28,37,54,82,62,89),(11,38,63,14,55,90,29,83),(12,15,30,39,56,84,64,91)], [(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)], [(1,56,57,12),(2,11,58,55),(3,54,59,10),(4,9,60,53),(5,52,49,8),(6,7,50,51),(13,75,82,18),(14,17,83,74),(15,73,84,16),(19,81,76,24),(20,23,77,80),(21,79,78,22),(25,70,71,36),(26,35,72,69),(27,68,61,34),(28,33,62,67),(29,66,63,32),(30,31,64,65),(37,94,89,42),(38,41,90,93),(39,92,91,40),(43,88,95,48),(44,47,96,87),(45,86,85,46)]])
Matrix representation of C8.2D12 ►in GL6(𝔽73)
| 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 34 | 5 | 39 | 68 |
| 0 | 0 | 68 | 39 | 5 | 34 |
| 0 | 0 | 34 | 5 | 34 | 5 |
| 0 | 0 | 68 | 39 | 68 | 39 |
| 72 | 71 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 15 | 27 | 23 |
| 0 | 0 | 58 | 58 | 50 | 50 |
| 0 | 0 | 27 | 23 | 30 | 58 |
| 0 | 0 | 50 | 50 | 15 | 15 |
| 72 | 71 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 58 | 58 | 50 | 50 |
| 0 | 0 | 43 | 15 | 27 | 23 |
| 0 | 0 | 50 | 50 | 15 | 15 |
| 0 | 0 | 27 | 23 | 30 | 58 |
G:=sub<GL(6,GF(73))| [72,0,0,0,0,0,0,72,0,0,0,0,0,0,34,68,34,68,0,0,5,39,5,39,0,0,39,5,34,68,0,0,68,34,5,39],[72,1,0,0,0,0,71,1,0,0,0,0,0,0,43,58,27,50,0,0,15,58,23,50,0,0,27,50,30,15,0,0,23,50,58,15],[72,0,0,0,0,0,71,1,0,0,0,0,0,0,58,43,50,27,0,0,58,15,50,23,0,0,50,27,15,30,0,0,50,23,15,58] >;
C8.2D12 in GAP, Magma, Sage, TeX
C_8._2D_{12} % in TeX
G:=Group("C8.2D12"); // GroupNames label
G:=SmallGroup(192,426);
// by ID
G=gap.SmallGroup(192,426);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,477,120,254,555,226,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^8=b^12=1,c^2=a^4,b*a*b^-1=a^3,c*a*c^-1=a^-1,c*b*c^-1=a^4*b^-1>;
// generators/relations
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