metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.22D4, C3⋊C8.19D4, (C6×D8).5C2, (C2×D8).4S3, C4.22(S3×D4), (C8×Dic3)⋊6C2, (C2×D4).63D6, (C2×C8).237D6, C6.33(C4○D8), C12.165(C2×D4), C8.15(C3⋊D4), C3⋊3(C8.12D4), (C2×Dic12)⋊18C2, C23.12D6⋊5C2, C6.28(C4⋊1D4), (C2×C24).89C22, (C6×D4).82C22, C22.255(S3×D4), C2.17(D8⋊3S3), C2.19(C12⋊3D4), (C2×C12).432C23, (C2×Dic3).111D4, (C2×Dic6).121C22, (C4×Dic3).239C22, C4.6(C2×C3⋊D4), (C2×D4.S3)⋊18C2, (C2×C6).345(C2×D4), (C2×C3⋊C8).270C22, (C2×C4).522(C22×S3), SmallGroup(192,714)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.22D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a17, cac-1=a-1, cbc-1=a12b-1 >
Subgroups: 376 in 130 conjugacy classes, 43 normal (21 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C2×C6, C2×C6, C42, C22⋊C4, C2×C8, C2×C8, D8, SD16, Q16, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, C2×Dic3, C2×Dic3, C2×C12, C3×D4, C22×C6, C4×C8, C4.4D4, C2×D8, C2×SD16, C2×Q16, Dic12, C2×C3⋊C8, C4×Dic3, D4.S3, C6.D4, C2×C24, C3×D8, C2×Dic6, C6×D4, C8.12D4, C8×Dic3, C2×Dic12, C2×D4.S3, C23.12D6, C6×D8, C24.22D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C3⋊D4, C22×S3, C4⋊1D4, C4○D8, S3×D4, C2×C3⋊D4, C8.12D4, D8⋊3S3, C12⋊3D4, C24.22D4
►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 37 49 74)(2 30 50 91)(3 47 51 84)(4 40 52 77)(5 33 53 94)(6 26 54 87)(7 43 55 80)(8 36 56 73)(9 29 57 90)(10 46 58 83)(11 39 59 76)(12 32 60 93)(13 25 61 86)(14 42 62 79)(15 35 63 96)(16 28 64 89)(17 45 65 82)(18 38 66 75)(19 31 67 92)(20 48 68 85)(21 41 69 78)(22 34 70 95)(23 27 71 88)(24 44 72 81)
(1 86 13 74)(2 85 14 73)(3 84 15 96)(4 83 16 95)(5 82 17 94)(6 81 18 93)(7 80 19 92)(8 79 20 91)(9 78 21 90)(10 77 22 89)(11 76 23 88)(12 75 24 87)(25 61 37 49)(26 60 38 72)(27 59 39 71)(28 58 40 70)(29 57 41 69)(30 56 42 68)(31 55 43 67)(32 54 44 66)(33 53 45 65)(34 52 46 64)(35 51 47 63)(36 50 48 62)
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), (1,37,49,74)(2,30,50,91)(3,47,51,84)(4,40,52,77)(5,33,53,94)(6,26,54,87)(7,43,55,80)(8,36,56,73)(9,29,57,90)(10,46,58,83)(11,39,59,76)(12,32,60,93)(13,25,61,86)(14,42,62,79)(15,35,63,96)(16,28,64,89)(17,45,65,82)(18,38,66,75)(19,31,67,92)(20,48,68,85)(21,41,69,78)(22,34,70,95)(23,27,71,88)(24,44,72,81), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,61,37,49)(26,60,38,72)(27,59,39,71)(28,58,40,70)(29,57,41,69)(30,56,42,68)(31,55,43,67)(32,54,44,66)(33,53,45,65)(34,52,46,64)(35,51,47,63)(36,50,48,62)>;
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), (1,37,49,74)(2,30,50,91)(3,47,51,84)(4,40,52,77)(5,33,53,94)(6,26,54,87)(7,43,55,80)(8,36,56,73)(9,29,57,90)(10,46,58,83)(11,39,59,76)(12,32,60,93)(13,25,61,86)(14,42,62,79)(15,35,63,96)(16,28,64,89)(17,45,65,82)(18,38,66,75)(19,31,67,92)(20,48,68,85)(21,41,69,78)(22,34,70,95)(23,27,71,88)(24,44,72,81), (1,86,13,74)(2,85,14,73)(3,84,15,96)(4,83,16,95)(5,82,17,94)(6,81,18,93)(7,80,19,92)(8,79,20,91)(9,78,21,90)(10,77,22,89)(11,76,23,88)(12,75,24,87)(25,61,37,49)(26,60,38,72)(27,59,39,71)(28,58,40,70)(29,57,41,69)(30,56,42,68)(31,55,43,67)(32,54,44,66)(33,53,45,65)(34,52,46,64)(35,51,47,63)(36,50,48,62) );
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)], [(1,37,49,74),(2,30,50,91),(3,47,51,84),(4,40,52,77),(5,33,53,94),(6,26,54,87),(7,43,55,80),(8,36,56,73),(9,29,57,90),(10,46,58,83),(11,39,59,76),(12,32,60,93),(13,25,61,86),(14,42,62,79),(15,35,63,96),(16,28,64,89),(17,45,65,82),(18,38,66,75),(19,31,67,92),(20,48,68,85),(21,41,69,78),(22,34,70,95),(23,27,71,88),(24,44,72,81)], [(1,86,13,74),(2,85,14,73),(3,84,15,96),(4,83,16,95),(5,82,17,94),(6,81,18,93),(7,80,19,92),(8,79,20,91),(9,78,21,90),(10,77,22,89),(11,76,23,88),(12,75,24,87),(25,61,37,49),(26,60,38,72),(27,59,39,71),(28,58,40,70),(29,57,41,69),(30,56,42,68),(31,55,43,67),(32,54,44,66),(33,53,45,65),(34,52,46,64),(35,51,47,63),(36,50,48,62)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 8 | 8 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 24 | 24 | 2 | 2 | 2 | 8 | 8 | 8 | 8 | 2 | 2 | 2 | 2 | 6 | 6 | 6 | 6 | 4 | 4 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | ||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C3⋊D4 | C4○D8 | S3×D4 | S3×D4 | D8⋊3S3 |
| kernel | C24.22D4 | C8×Dic3 | C2×Dic12 | C2×D4.S3 | C23.12D6 | C6×D8 | C2×D8 | C3⋊C8 | C24 | C2×Dic3 | C2×C8 | C2×D4 | C8 | C6 | C4 | C22 | C2 |
| # reps | 1 | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 1 | 2 | 4 | 8 | 1 | 1 | 4 |
Matrix representation of C24.22D4 ►in GL4(𝔽73) generated by
| 64 | 0 | 0 | 0 |
| 8 | 8 | 0 | 0 |
| 0 | 0 | 32 | 32 |
| 0 | 0 | 57 | 0 |
| 72 | 7 | 0 | 0 |
| 31 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 0 | 46 |
| 72 | 7 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 46 | 0 |
| 0 | 0 | 27 | 27 |
G:=sub<GL(4,GF(73))| [64,8,0,0,0,8,0,0,0,0,32,57,0,0,32,0],[72,31,0,0,7,1,0,0,0,0,46,0,0,0,0,46],[72,0,0,0,7,1,0,0,0,0,46,27,0,0,0,27] >;
C24.22D4 in GAP, Magma, Sage, TeX
C_{24}._{22}D_4 % in TeX
G:=Group("C24.22D4"); // GroupNames label
G:=SmallGroup(192,714);
// by ID
G=gap.SmallGroup(192,714);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,112,701,1094,135,570,297,136,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^17,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations