metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C12.57D8, C12.41SD16, C42.194D6, C12.7M4(2), D4:(C3:C8), C3:3(D4:C8), (C3xD4):1C8, C6.14C4wrC2, C12.7(C2xC8), C12:C8:9C2, (C6xD4).7C4, (C4xD4).1S3, (D4xC12).1C2, C4:C4.3Dic3, C4.30(D4:S3), (C2xC12).489D4, (C2xD4).4Dic3, C6.15(C22:C8), C4.16(D4.S3), (C4xC12).45C22, C4.1(C4.Dic3), C6.22(D4:C4), C2.2(D4:Dic3), C2.2(Q8:3Dic3), C2.5(C12.55D4), C22.29(C6.D4), (C4xC3:C8):2C2, C4.1(C2xC3:C8), (C3xC4:C4).5C4, (C2xC12).59(C2xC4), (C2xC4).37(C2xDic3), (C2xC4).161(C3:D4), (C2xC6).92(C22:C4), SmallGroup(192,93)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C12.57D8
G = < a,b,c | a12=b8=1, c2=a9, bab-1=cac-1=a5, cbc-1=a9b-1 >
Subgroups: 184 in 82 conjugacy classes, 39 normal (35 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, C6, C6, C8, C2xC4, C2xC4, D4, D4, C23, C12, C12, C2xC6, C2xC6, C42, C22:C4, C4:C4, C2xC8, C22xC4, C2xD4, C3:C8, C2xC12, C2xC12, C3xD4, C3xD4, C22xC6, C4xC8, C4:C8, C4xD4, C2xC3:C8, C4xC12, C3xC22:C4, C3xC4:C4, C22xC12, C6xD4, D4:C8, C4xC3:C8, C12:C8, D4xC12, C12.57D8
Quotients: C1, C2, C4, C22, S3, C8, C2xC4, D4, Dic3, D6, C22:C4, C2xC8, M4(2), D8, SD16, C3:C8, C2xDic3, C3:D4, C22:C8, D4:C4, C4wrC2, C2xC3:C8, C4.Dic3, D4:S3, D4.S3, C6.D4, D4:C8, C12.55D4, D4:Dic3, Q8:3Dic3, C12.57D8
►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 54 46 69 85 74 27 13)(2 59 47 62 86 79 28 18)(3 52 48 67 87 84 29 23)(4 57 37 72 88 77 30 16)(5 50 38 65 89 82 31 21)(6 55 39 70 90 75 32 14)(7 60 40 63 91 80 33 19)(8 53 41 68 92 73 34 24)(9 58 42 61 93 78 35 17)(10 51 43 66 94 83 36 22)(11 56 44 71 95 76 25 15)(12 49 45 64 96 81 26 20)
(1 22 10 19 7 16 4 13)(2 15 11 24 8 21 5 18)(3 20 12 17 9 14 6 23)(25 59 34 56 31 53 28 50)(26 52 35 49 32 58 29 55)(27 57 36 54 33 51 30 60)(37 80 46 77 43 74 40 83)(38 73 47 82 44 79 41 76)(39 78 48 75 45 84 42 81)(61 93 70 90 67 87 64 96)(62 86 71 95 68 92 65 89)(63 91 72 88 69 85 66 94)
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), (1,54,46,69,85,74,27,13)(2,59,47,62,86,79,28,18)(3,52,48,67,87,84,29,23)(4,57,37,72,88,77,30,16)(5,50,38,65,89,82,31,21)(6,55,39,70,90,75,32,14)(7,60,40,63,91,80,33,19)(8,53,41,68,92,73,34,24)(9,58,42,61,93,78,35,17)(10,51,43,66,94,83,36,22)(11,56,44,71,95,76,25,15)(12,49,45,64,96,81,26,20), (1,22,10,19,7,16,4,13)(2,15,11,24,8,21,5,18)(3,20,12,17,9,14,6,23)(25,59,34,56,31,53,28,50)(26,52,35,49,32,58,29,55)(27,57,36,54,33,51,30,60)(37,80,46,77,43,74,40,83)(38,73,47,82,44,79,41,76)(39,78,48,75,45,84,42,81)(61,93,70,90,67,87,64,96)(62,86,71,95,68,92,65,89)(63,91,72,88,69,85,66,94)>;
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), (1,54,46,69,85,74,27,13)(2,59,47,62,86,79,28,18)(3,52,48,67,87,84,29,23)(4,57,37,72,88,77,30,16)(5,50,38,65,89,82,31,21)(6,55,39,70,90,75,32,14)(7,60,40,63,91,80,33,19)(8,53,41,68,92,73,34,24)(9,58,42,61,93,78,35,17)(10,51,43,66,94,83,36,22)(11,56,44,71,95,76,25,15)(12,49,45,64,96,81,26,20), (1,22,10,19,7,16,4,13)(2,15,11,24,8,21,5,18)(3,20,12,17,9,14,6,23)(25,59,34,56,31,53,28,50)(26,52,35,49,32,58,29,55)(27,57,36,54,33,51,30,60)(37,80,46,77,43,74,40,83)(38,73,47,82,44,79,41,76)(39,78,48,75,45,84,42,81)(61,93,70,90,67,87,64,96)(62,86,71,95,68,92,65,89)(63,91,72,88,69,85,66,94) );
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)], [(1,54,46,69,85,74,27,13),(2,59,47,62,86,79,28,18),(3,52,48,67,87,84,29,23),(4,57,37,72,88,77,30,16),(5,50,38,65,89,82,31,21),(6,55,39,70,90,75,32,14),(7,60,40,63,91,80,33,19),(8,53,41,68,92,73,34,24),(9,58,42,61,93,78,35,17),(10,51,43,66,94,83,36,22),(11,56,44,71,95,76,25,15),(12,49,45,64,96,81,26,20)], [(1,22,10,19,7,16,4,13),(2,15,11,24,8,21,5,18),(3,20,12,17,9,14,6,23),(25,59,34,56,31,53,28,50),(26,52,35,49,32,58,29,55),(27,57,36,54,33,51,30,60),(37,80,46,77,43,74,40,83),(38,73,47,82,44,79,41,76),(39,78,48,75,45,84,42,81),(61,93,70,90,67,87,64,96),(62,86,71,95,68,92,65,89),(63,91,72,88,69,85,66,94)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | ··· | 8H | 8I | 8J | 8K | 8L | 12A | 12B | 12C | 12D | 12E | ··· | 12L |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | ··· | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 |
| size | 1 | 1 | 1 | 1 | 4 | 4 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 4 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | - | - | + | + | - | ||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | S3 | D4 | D6 | Dic3 | Dic3 | M4(2) | D8 | SD16 | C3:D4 | C3:C8 | C4wrC2 | C4.Dic3 | D4:S3 | D4.S3 | Q8:3Dic3 |
| kernel | C12.57D8 | C4xC3:C8 | C12:C8 | D4xC12 | C3xC4:C4 | C6xD4 | C3xD4 | C4xD4 | C2xC12 | C42 | C4:C4 | C2xD4 | C12 | C12 | C12 | C2xC4 | D4 | C6 | C4 | C4 | C4 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 8 | 1 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 1 | 1 | 2 |
Matrix representation of C12.57D8 ►in GL4(F73) generated by
| 24 | 58 | 0 | 0 |
| 0 | 3 | 0 | 0 |
| 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 27 |
| 24 | 71 | 0 | 0 |
| 19 | 49 | 0 | 0 |
| 0 | 0 | 59 | 14 |
| 0 | 0 | 59 | 59 |
| 49 | 1 | 0 | 0 |
| 54 | 24 | 0 | 0 |
| 0 | 0 | 59 | 14 |
| 0 | 0 | 14 | 14 |
G:=sub<GL(4,GF(73))| [24,0,0,0,58,3,0,0,0,0,27,0,0,0,0,27],[24,19,0,0,71,49,0,0,0,0,59,59,0,0,14,59],[49,54,0,0,1,24,0,0,0,0,59,14,0,0,14,14] >;
C12.57D8 in GAP, Magma, Sage, TeX
C_{12}._{57}D_8 % in TeX
G:=Group("C12.57D8"); // GroupNames label
G:=SmallGroup(192,93);
// by ID
G=gap.SmallGroup(192,93);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,28,141,100,1123,570,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^12=b^8=1,c^2=a^9,b*a*b^-1=c*a*c^-1=a^5,c*b*c^-1=a^9*b^-1>;
// generators/relations