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, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, M5(2), C3×Dic3, C3×C12, S3×C6, C3⋊C16, C3⋊C16, 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, C4, C22, S3, C8, C2×C4, Dic3, D6, C2×C8, C3⋊C8, 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 65 17 49 9 57)(2 58 18 66 10 50)(3 51 19 59 11 67)(4 68 20 52 12 60)(5 61 21 69 13 53)(6 54 22 62 14 70)(7 71 23 55 15 63)(8 64 24 72 16 56)(25 93 33 85 41 77)(26 86 34 78 42 94)(27 79 35 95 43 87)(28 96 36 88 44 80)(29 89 37 81 45 73)(30 82 38 74 46 90)(31 75 39 91 47 83)(32 92 40 84 48 76)
(1 95 22 92 19 89 16 86 13 83 10 80 7 77 4 74)(2 88 23 85 20 82 17 79 14 76 11 73 8 94 5 91)(3 81 24 78 21 75 18 96 15 93 12 90 9 87 6 84)(25 72 46 69 43 66 40 63 37 60 34 57 31 54 28 51)(26 65 47 62 44 59 41 56 38 53 35 50 32 71 29 68)(27 58 48 55 45 52 42 49 39 70 36 67 33 64 30 61)
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), (1,65,17,49,9,57)(2,58,18,66,10,50)(3,51,19,59,11,67)(4,68,20,52,12,60)(5,61,21,69,13,53)(6,54,22,62,14,70)(7,71,23,55,15,63)(8,64,24,72,16,56)(25,93,33,85,41,77)(26,86,34,78,42,94)(27,79,35,95,43,87)(28,96,36,88,44,80)(29,89,37,81,45,73)(30,82,38,74,46,90)(31,75,39,91,47,83)(32,92,40,84,48,76), (1,95,22,92,19,89,16,86,13,83,10,80,7,77,4,74)(2,88,23,85,20,82,17,79,14,76,11,73,8,94,5,91)(3,81,24,78,21,75,18,96,15,93,12,90,9,87,6,84)(25,72,46,69,43,66,40,63,37,60,34,57,31,54,28,51)(26,65,47,62,44,59,41,56,38,53,35,50,32,71,29,68)(27,58,48,55,45,52,42,49,39,70,36,67,33,64,30,61)>;
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), (1,65,17,49,9,57)(2,58,18,66,10,50)(3,51,19,59,11,67)(4,68,20,52,12,60)(5,61,21,69,13,53)(6,54,22,62,14,70)(7,71,23,55,15,63)(8,64,24,72,16,56)(25,93,33,85,41,77)(26,86,34,78,42,94)(27,79,35,95,43,87)(28,96,36,88,44,80)(29,89,37,81,45,73)(30,82,38,74,46,90)(31,75,39,91,47,83)(32,92,40,84,48,76), (1,95,22,92,19,89,16,86,13,83,10,80,7,77,4,74)(2,88,23,85,20,82,17,79,14,76,11,73,8,94,5,91)(3,81,24,78,21,75,18,96,15,93,12,90,9,87,6,84)(25,72,46,69,43,66,40,63,37,60,34,57,31,54,28,51)(26,65,47,62,44,59,41,56,38,53,35,50,32,71,29,68)(27,58,48,55,45,52,42,49,39,70,36,67,33,64,30,61) );
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)], [(1,65,17,49,9,57),(2,58,18,66,10,50),(3,51,19,59,11,67),(4,68,20,52,12,60),(5,61,21,69,13,53),(6,54,22,62,14,70),(7,71,23,55,15,63),(8,64,24,72,16,56),(25,93,33,85,41,77),(26,86,34,78,42,94),(27,79,35,95,43,87),(28,96,36,88,44,80),(29,89,37,81,45,73),(30,82,38,74,46,90),(31,75,39,91,47,83),(32,92,40,84,48,76)], [(1,95,22,92,19,89,16,86,13,83,10,80,7,77,4,74),(2,88,23,85,20,82,17,79,14,76,11,73,8,94,5,91),(3,81,24,78,21,75,18,96,15,93,12,90,9,87,6,84),(25,72,46,69,43,66,40,63,37,60,34,57,31,54,28,51),(26,65,47,62,44,59,41,56,38,53,35,50,32,71,29,68),(27,58,48,55,45,52,42,49,39,70,36,67,33,64,30,61)]])
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