metabelian, supersoluble, monomial
Aliases: C24.61D6, C32⋊4M5(2), C3⋊C16⋊4S3, C8.22S32, D6.(C3⋊C8), (S3×C8).2S3, (S3×C6).2C8, C6.18(S3×C8), Dic3.(C3⋊C8), C3⋊C8.2Dic3, (S3×C24).4C2, (S3×C12).1C4, C12.95(C4×S3), C3⋊3(D6.C8), C24.S3⋊7C2, (C4×S3).2Dic3, (C3×Dic3).1C8, C4.17(S3×Dic3), C3⋊1(C12.C8), (C3×C24).43C22, C12.28(C2×Dic3), C6.2(C2×C3⋊C8), C2.3(S3×C3⋊C8), (C3×C3⋊C16)⋊9C2, (C3×C3⋊C8).4C4, (C3×C6).14(C2×C8), (C3×C12).79(C2×C4), SmallGroup(288,191)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.61D6
G = < a,b,c | a24=b6=1, c2=a21, bab-1=cac-1=a17, cbc-1=a12b-1 >
Subgroups: 126 in 57 conjugacy classes, 30 normal (all characteristic)
C1, C2, C2, C3 [×2], C3, C4, C4, C22, S3, C6 [×2], C6 [×2], C8, C8, C2×C4, C32, Dic3, C12 [×2], C12 [×2], D6, C2×C6, C16 [×2], C2×C8, C3×S3, C3×C6, C3⋊C8, C24 [×2], C24 [×2], C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C3⋊C16 [×3], C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, D6.C8, C12.C8, C3×C3⋊C16, C24.S3, S3×C24, C24.61D6
Quotients: C1, C2 [×3], C4 [×2], C22, S3 [×2], C8 [×2], C2×C4, Dic3 [×2], D6 [×2], C2×C8, C3⋊C8 [×2], C4×S3, C2×Dic3, M5(2), S32, S3×C8, C2×C3⋊C8, S3×Dic3, D6.C8, C12.C8, S3×C3⋊C8, C24.61D6
►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 53 17 61 9 69)(2 70 18 54 10 62)(3 63 19 71 11 55)(4 56 20 64 12 72)(5 49 21 57 13 65)(6 66 22 50 14 58)(7 59 23 67 15 51)(8 52 24 60 16 68)(25 88 33 80 41 96)(26 81 34 73 42 89)(27 74 35 90 43 82)(28 91 36 83 44 75)(29 84 37 76 45 92)(30 77 38 93 46 85)(31 94 39 86 47 78)(32 87 40 79 48 95)
(1 96 22 93 19 90 16 87 13 84 10 81 7 78 4 75)(2 89 23 86 20 83 17 80 14 77 11 74 8 95 5 92)(3 82 24 79 21 76 18 73 15 94 12 91 9 88 6 85)(25 54 46 51 43 72 40 69 37 66 34 63 31 60 28 57)(26 71 47 68 44 65 41 62 38 59 35 56 32 53 29 50)(27 64 48 61 45 58 42 55 39 52 36 49 33 70 30 67)
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), (1,53,17,61,9,69)(2,70,18,54,10,62)(3,63,19,71,11,55)(4,56,20,64,12,72)(5,49,21,57,13,65)(6,66,22,50,14,58)(7,59,23,67,15,51)(8,52,24,60,16,68)(25,88,33,80,41,96)(26,81,34,73,42,89)(27,74,35,90,43,82)(28,91,36,83,44,75)(29,84,37,76,45,92)(30,77,38,93,46,85)(31,94,39,86,47,78)(32,87,40,79,48,95), (1,96,22,93,19,90,16,87,13,84,10,81,7,78,4,75)(2,89,23,86,20,83,17,80,14,77,11,74,8,95,5,92)(3,82,24,79,21,76,18,73,15,94,12,91,9,88,6,85)(25,54,46,51,43,72,40,69,37,66,34,63,31,60,28,57)(26,71,47,68,44,65,41,62,38,59,35,56,32,53,29,50)(27,64,48,61,45,58,42,55,39,52,36,49,33,70,30,67)>;
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), (1,53,17,61,9,69)(2,70,18,54,10,62)(3,63,19,71,11,55)(4,56,20,64,12,72)(5,49,21,57,13,65)(6,66,22,50,14,58)(7,59,23,67,15,51)(8,52,24,60,16,68)(25,88,33,80,41,96)(26,81,34,73,42,89)(27,74,35,90,43,82)(28,91,36,83,44,75)(29,84,37,76,45,92)(30,77,38,93,46,85)(31,94,39,86,47,78)(32,87,40,79,48,95), (1,96,22,93,19,90,16,87,13,84,10,81,7,78,4,75)(2,89,23,86,20,83,17,80,14,77,11,74,8,95,5,92)(3,82,24,79,21,76,18,73,15,94,12,91,9,88,6,85)(25,54,46,51,43,72,40,69,37,66,34,63,31,60,28,57)(26,71,47,68,44,65,41,62,38,59,35,56,32,53,29,50)(27,64,48,61,45,58,42,55,39,52,36,49,33,70,30,67) );
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)], [(1,53,17,61,9,69),(2,70,18,54,10,62),(3,63,19,71,11,55),(4,56,20,64,12,72),(5,49,21,57,13,65),(6,66,22,50,14,58),(7,59,23,67,15,51),(8,52,24,60,16,68),(25,88,33,80,41,96),(26,81,34,73,42,89),(27,74,35,90,43,82),(28,91,36,83,44,75),(29,84,37,76,45,92),(30,77,38,93,46,85),(31,94,39,86,47,78),(32,87,40,79,48,95)], [(1,96,22,93,19,90,16,87,13,84,10,81,7,78,4,75),(2,89,23,86,20,83,17,80,14,77,11,74,8,95,5,92),(3,82,24,79,21,76,18,73,15,94,12,91,9,88,6,85),(25,54,46,51,43,72,40,69,37,66,34,63,31,60,28,57),(26,71,47,68,44,65,41,62,38,59,35,56,32,53,29,50),(27,64,48,61,45,58,42,55,39,52,36,49,33,70,30,67)])
60 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 4A | 4B | 4C | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | 24A | ··· | 24H | 24I | 24J | 24K | 24L | 24M | 24N | 24O | 24P | 48A | ··· | 48H |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 6 | 2 | 2 | 4 | 1 | 1 | 6 | 2 | 2 | 4 | 6 | 6 | 1 | 1 | 1 | 1 | 6 | 6 | 2 | 2 | 2 | 2 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 18 | 18 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | ··· | 6 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | - | + | - | + | - | |||||||||||||
| image | C1 | C2 | C2 | C2 | C4 | C4 | C8 | C8 | S3 | S3 | Dic3 | D6 | Dic3 | C3⋊C8 | C4×S3 | C3⋊C8 | M5(2) | S3×C8 | D6.C8 | C12.C8 | S32 | S3×Dic3 | S3×C3⋊C8 | C24.61D6 |
| kernel | C24.61D6 | C3×C3⋊C16 | C24.S3 | S3×C24 | C3×C3⋊C8 | S3×C12 | C3×Dic3 | S3×C6 | C3⋊C16 | S3×C8 | C3⋊C8 | C24 | C4×S3 | Dic3 | C12 | D6 | C32 | C6 | C3 | C3 | C8 | C4 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 4 | 4 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 1 | 1 | 2 | 4 |
Matrix representation of C24.61D6 ►in GL6(𝔽97)
| 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 47 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 22 | 0 | 0 |
| 0 | 0 | 75 | 75 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 95 | 80 | 0 | 0 | 0 | 0 |
| 23 | 2 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 96 | 96 |
| 74 | 95 | 0 | 0 | 0 | 0 |
| 94 | 23 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 33 | 0 | 0 |
| 0 | 0 | 33 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 1 | 0 |
G:=sub<GL(6,GF(97))| [47,0,0,0,0,0,0,47,0,0,0,0,0,0,0,75,0,0,0,0,22,75,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[95,23,0,0,0,0,80,2,0,0,0,0,0,0,0,96,0,0,0,0,96,0,0,0,0,0,0,0,0,96,0,0,0,0,1,96],[74,94,0,0,0,0,95,23,0,0,0,0,0,0,0,33,0,0,0,0,33,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0] >;
C24.61D6 in GAP, Magma, Sage, TeX
C_{24}._{61}D_6 % in TeX
G:=Group("C24.61D6"); // GroupNames label
G:=SmallGroup(288,191);
// by ID
G=gap.SmallGroup(288,191);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,253,36,58,80,1356,9414]);
// Polycyclic
G:=Group<a,b,c|a^24=b^6=1,c^2=a^21,b*a*b^-1=c*a*c^-1=a^17,c*b*c^-1=a^12*b^-1>;
// generators/relations