metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.9D8, C42.9D6, C12.8SD16, (C6×D4).3C4, C12⋊C8⋊10C2, C4⋊1D4.2S3, C3⋊2(C4.D8), C4.12(D4⋊S3), (C2×C12).106D4, C4.6(D4.S3), (C2×D4).3Dic3, (C4×C12).47C22, C6.9(C4.D4), C6.24(D4⋊C4), C2.4(C12.D4), C2.4(D4⋊Dic3), C22.41(C6.D4), (C3×C4⋊1D4).1C2, (C2×C12).171(C2×C4), (C2×C4).11(C2×Dic3), (C2×C4).176(C3⋊D4), (C2×C6).102(C22⋊C4), SmallGroup(192,103)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.9D8
G = < a,b,c | a12=b8=1, c2=a9, bab-1=a-1, cac-1=a5, cbc-1=a9b-1 >
Subgroups: 240 in 84 conjugacy classes, 35 normal (13 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C2×C4, C2×C4, D4, C23, C12, C12, C2×C6, C2×C6, C42, C2×C8, C2×D4, C2×D4, C3⋊C8, C2×C12, C2×C12, C3×D4, C22×C6, C4⋊C8, C4⋊1D4, C2×C3⋊C8, C4×C12, C6×D4, C6×D4, C4.D8, C12⋊C8, C3×C4⋊1D4, C12.9D8
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, Dic3, D6, C22⋊C4, D8, SD16, C2×Dic3, C3⋊D4, C4.D4, D4⋊C4, D4⋊S3, D4.S3, C6.D4, C4.D8, D4⋊Dic3, C12.D4, C12.9D8
►On 96 points
Generators in S96
(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 49 16 40 66 94 26 82)(2 60 17 39 67 93 27 81)(3 59 18 38 68 92 28 80)(4 58 19 37 69 91 29 79)(5 57 20 48 70 90 30 78)(6 56 21 47 71 89 31 77)(7 55 22 46 72 88 32 76)(8 54 23 45 61 87 33 75)(9 53 24 44 62 86 34 74)(10 52 13 43 63 85 35 73)(11 51 14 42 64 96 36 84)(12 50 15 41 65 95 25 83)
(1 79 10 76 7 73 4 82)(2 84 11 81 8 78 5 75)(3 77 12 74 9 83 6 80)(13 88 22 85 19 94 16 91)(14 93 23 90 20 87 17 96)(15 86 24 95 21 92 18 89)(25 53 34 50 31 59 28 56)(26 58 35 55 32 52 29 49)(27 51 36 60 33 57 30 54)(37 63 46 72 43 69 40 66)(38 68 47 65 44 62 41 71)(39 61 48 70 45 67 42 64)
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), (1,49,16,40,66,94,26,82)(2,60,17,39,67,93,27,81)(3,59,18,38,68,92,28,80)(4,58,19,37,69,91,29,79)(5,57,20,48,70,90,30,78)(6,56,21,47,71,89,31,77)(7,55,22,46,72,88,32,76)(8,54,23,45,61,87,33,75)(9,53,24,44,62,86,34,74)(10,52,13,43,63,85,35,73)(11,51,14,42,64,96,36,84)(12,50,15,41,65,95,25,83), (1,79,10,76,7,73,4,82)(2,84,11,81,8,78,5,75)(3,77,12,74,9,83,6,80)(13,88,22,85,19,94,16,91)(14,93,23,90,20,87,17,96)(15,86,24,95,21,92,18,89)(25,53,34,50,31,59,28,56)(26,58,35,55,32,52,29,49)(27,51,36,60,33,57,30,54)(37,63,46,72,43,69,40,66)(38,68,47,65,44,62,41,71)(39,61,48,70,45,67,42,64)>;
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), (1,49,16,40,66,94,26,82)(2,60,17,39,67,93,27,81)(3,59,18,38,68,92,28,80)(4,58,19,37,69,91,29,79)(5,57,20,48,70,90,30,78)(6,56,21,47,71,89,31,77)(7,55,22,46,72,88,32,76)(8,54,23,45,61,87,33,75)(9,53,24,44,62,86,34,74)(10,52,13,43,63,85,35,73)(11,51,14,42,64,96,36,84)(12,50,15,41,65,95,25,83), (1,79,10,76,7,73,4,82)(2,84,11,81,8,78,5,75)(3,77,12,74,9,83,6,80)(13,88,22,85,19,94,16,91)(14,93,23,90,20,87,17,96)(15,86,24,95,21,92,18,89)(25,53,34,50,31,59,28,56)(26,58,35,55,32,52,29,49)(27,51,36,60,33,57,30,54)(37,63,46,72,43,69,40,66)(38,68,47,65,44,62,41,71)(39,61,48,70,45,67,42,64) );
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)], [(1,49,16,40,66,94,26,82),(2,60,17,39,67,93,27,81),(3,59,18,38,68,92,28,80),(4,58,19,37,69,91,29,79),(5,57,20,48,70,90,30,78),(6,56,21,47,71,89,31,77),(7,55,22,46,72,88,32,76),(8,54,23,45,61,87,33,75),(9,53,24,44,62,86,34,74),(10,52,13,43,63,85,35,73),(11,51,14,42,64,96,36,84),(12,50,15,41,65,95,25,83)], [(1,79,10,76,7,73,4,82),(2,84,11,81,8,78,5,75),(3,77,12,74,9,83,6,80),(13,88,22,85,19,94,16,91),(14,93,23,90,20,87,17,96),(15,86,24,95,21,92,18,89),(25,53,34,50,31,59,28,56),(26,58,35,55,32,52,29,49),(27,51,36,60,33,57,30,54),(37,63,46,72,43,69,40,66),(38,68,47,65,44,62,41,71),(39,61,48,70,45,67,42,64)]])
33 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 12A | ··· | 12F |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 4 | ··· | 4 |
33 irreducible representations
| dim | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | + | + | - | ||||
| image | C1 | C2 | C2 | C4 | S3 | D4 | D6 | Dic3 | D8 | SD16 | C3⋊D4 | C4.D4 | D4⋊S3 | D4.S3 | C12.D4 |
| kernel | C12.9D8 | C12⋊C8 | C3×C4⋊1D4 | C6×D4 | C4⋊1D4 | C2×C12 | C42 | C2×D4 | C12 | C12 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 2 | 1 | 4 | 1 | 2 | 1 | 2 | 4 | 4 | 4 | 1 | 2 | 2 | 2 |
Matrix representation of C12.9D8 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 65 | 0 |
| 0 | 0 | 0 | 0 | 25 | 9 |
| 57 | 16 | 0 | 0 | 0 | 0 |
| 16 | 16 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 16 | 57 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 2 |
| 0 | 0 | 0 | 0 | 40 | 40 |
| 57 | 16 | 0 | 0 | 0 | 0 |
| 57 | 57 | 0 | 0 | 0 | 0 |
| 0 | 0 | 57 | 57 | 0 | 0 |
| 0 | 0 | 57 | 16 | 0 | 0 |
| 0 | 0 | 0 | 0 | 33 | 2 |
| 0 | 0 | 0 | 0 | 39 | 40 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,65,25,0,0,0,0,0,9],[57,16,0,0,0,0,16,16,0,0,0,0,0,0,57,16,0,0,0,0,57,57,0,0,0,0,0,0,33,40,0,0,0,0,2,40],[57,57,0,0,0,0,16,57,0,0,0,0,0,0,57,57,0,0,0,0,57,16,0,0,0,0,0,0,33,39,0,0,0,0,2,40] >;
C12.9D8 in GAP, Magma, Sage, TeX
C_{12}._9D_8 % in TeX
G:=Group("C12.9D8"); // GroupNames label
G:=SmallGroup(192,103);
// by ID
G=gap.SmallGroup(192,103);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,219,100,1571,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=a^-1,c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations