metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24.4D4, C23.24D12, C8⋊Dic3⋊5C2, (C2×C8).79D6, (C2×C4).54D12, C8.1(C3⋊D4), C3⋊4(C8.D4), C12.423(C2×D4), (C2×C12).299D4, (C2×Dic12)⋊12C2, C6.75(C4⋊D4), (C2×C24).65C22, C2.Dic12⋊42C2, (C22×C6).105D4, (C22×C4).161D6, (C2×M4(2)).3S3, (C6×M4(2)).3C2, C12.232(C4○D4), C4.116(C4○D12), C2.23(C12⋊7D4), (C2×C12).777C23, C2.23(C8.D6), C22.136(C2×D12), C6.23(C8.C22), C4⋊Dic3.28C22, C12.48D4.17C2, (C2×Dic6).20C22, (C22×C12).306C22, (C2×C6).167(C2×D4), C4.116(C2×C3⋊D4), (C2×C4).726(C22×S3), SmallGroup(192,696)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24.4D4
G = < a,b,c | a24=b4=1, c2=a12, bab-1=a11, cac-1=a-1, cbc-1=a12b-1 >
Subgroups: 296 in 110 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, C6, C6, C6, C8, C8, C2×C4, C2×C4, Q8, C23, Dic3, C12, C12, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, M4(2), Q16, C22×C4, C2×Q8, C24, C24, Dic6, C2×Dic3, C2×C12, C2×C12, C22×C6, Q8⋊C4, C4.Q8, C22⋊Q8, C2×M4(2), C2×Q16, Dic12, Dic3⋊C4, C4⋊Dic3, C6.D4, C2×C24, C3×M4(2), C2×Dic6, C22×C12, C8.D4, C2.Dic12, C8⋊Dic3, C2×Dic12, C12.48D4, C6×M4(2), C24.4D4
Quotients: C1, C2, C22, S3, D4, C23, D6, C2×D4, C4○D4, D12, C3⋊D4, C22×S3, C4⋊D4, C8.C22, C2×D12, C4○D12, C2×C3⋊D4, C8.D4, C8.D6, C12⋊7D4, C24.4D4
►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 40 54 82)(2 27 55 93)(3 38 56 80)(4 25 57 91)(5 36 58 78)(6 47 59 89)(7 34 60 76)(8 45 61 87)(9 32 62 74)(10 43 63 85)(11 30 64 96)(12 41 65 83)(13 28 66 94)(14 39 67 81)(15 26 68 92)(16 37 69 79)(17 48 70 90)(18 35 71 77)(19 46 72 88)(20 33 49 75)(21 44 50 86)(22 31 51 73)(23 42 52 84)(24 29 53 95)
(1 94 13 82)(2 93 14 81)(3 92 15 80)(4 91 16 79)(5 90 17 78)(6 89 18 77)(7 88 19 76)(8 87 20 75)(9 86 21 74)(10 85 22 73)(11 84 23 96)(12 83 24 95)(25 69 37 57)(26 68 38 56)(27 67 39 55)(28 66 40 54)(29 65 41 53)(30 64 42 52)(31 63 43 51)(32 62 44 50)(33 61 45 49)(34 60 46 72)(35 59 47 71)(36 58 48 70)
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), (1,40,54,82)(2,27,55,93)(3,38,56,80)(4,25,57,91)(5,36,58,78)(6,47,59,89)(7,34,60,76)(8,45,61,87)(9,32,62,74)(10,43,63,85)(11,30,64,96)(12,41,65,83)(13,28,66,94)(14,39,67,81)(15,26,68,92)(16,37,69,79)(17,48,70,90)(18,35,71,77)(19,46,72,88)(20,33,49,75)(21,44,50,86)(22,31,51,73)(23,42,52,84)(24,29,53,95), (1,94,13,82)(2,93,14,81)(3,92,15,80)(4,91,16,79)(5,90,17,78)(6,89,18,77)(7,88,19,76)(8,87,20,75)(9,86,21,74)(10,85,22,73)(11,84,23,96)(12,83,24,95)(25,69,37,57)(26,68,38,56)(27,67,39,55)(28,66,40,54)(29,65,41,53)(30,64,42,52)(31,63,43,51)(32,62,44,50)(33,61,45,49)(34,60,46,72)(35,59,47,71)(36,58,48,70)>;
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), (1,40,54,82)(2,27,55,93)(3,38,56,80)(4,25,57,91)(5,36,58,78)(6,47,59,89)(7,34,60,76)(8,45,61,87)(9,32,62,74)(10,43,63,85)(11,30,64,96)(12,41,65,83)(13,28,66,94)(14,39,67,81)(15,26,68,92)(16,37,69,79)(17,48,70,90)(18,35,71,77)(19,46,72,88)(20,33,49,75)(21,44,50,86)(22,31,51,73)(23,42,52,84)(24,29,53,95), (1,94,13,82)(2,93,14,81)(3,92,15,80)(4,91,16,79)(5,90,17,78)(6,89,18,77)(7,88,19,76)(8,87,20,75)(9,86,21,74)(10,85,22,73)(11,84,23,96)(12,83,24,95)(25,69,37,57)(26,68,38,56)(27,67,39,55)(28,66,40,54)(29,65,41,53)(30,64,42,52)(31,63,43,51)(32,62,44,50)(33,61,45,49)(34,60,46,72)(35,59,47,71)(36,58,48,70) );
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)], [(1,40,54,82),(2,27,55,93),(3,38,56,80),(4,25,57,91),(5,36,58,78),(6,47,59,89),(7,34,60,76),(8,45,61,87),(9,32,62,74),(10,43,63,85),(11,30,64,96),(12,41,65,83),(13,28,66,94),(14,39,67,81),(15,26,68,92),(16,37,69,79),(17,48,70,90),(18,35,71,77),(19,46,72,88),(20,33,49,75),(21,44,50,86),(22,31,51,73),(23,42,52,84),(24,29,53,95)], [(1,94,13,82),(2,93,14,81),(3,92,15,80),(4,91,16,79),(5,90,17,78),(6,89,18,77),(7,88,19,76),(8,87,20,75),(9,86,21,74),(10,85,22,73),(11,84,23,96),(12,83,24,95),(25,69,37,57),(26,68,38,56),(27,67,39,55),(28,66,40,54),(29,65,41,53),(30,64,42,52),(31,63,43,51),(32,62,44,50),(33,61,45,49),(34,60,46,72),(35,59,47,71),(36,58,48,70)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 4 | 2 | 2 | 2 | 4 | 24 | 24 | 24 | 24 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | ··· | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | - | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | S3 | D4 | D4 | D4 | D6 | D6 | C4○D4 | C3⋊D4 | D12 | D12 | C4○D12 | C8.C22 | C8.D6 |
| kernel | C24.4D4 | C2.Dic12 | C8⋊Dic3 | C2×Dic12 | C12.48D4 | C6×M4(2) | C2×M4(2) | C24 | C2×C12 | C22×C6 | C2×C8 | C22×C4 | C12 | C8 | C2×C4 | C23 | C4 | C6 | C2 |
| # reps | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C24.4D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 66 | 7 | 0 | 0 |
| 0 | 0 | 66 | 59 | 0 | 0 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 54 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 30 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 46 | 46 |
| 0 | 0 | 0 | 0 | 0 | 27 |
| 24 | 15 | 0 | 0 | 0 | 0 |
| 59 | 49 | 0 | 0 | 0 | 0 |
| 0 | 0 | 43 | 30 | 0 | 0 |
| 0 | 0 | 60 | 30 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 27 |
| 0 | 0 | 0 | 0 | 0 | 46 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,66,66,0,0,0,0,7,59,0,0,72,1,0,0,0,0,72,0,0,0],[24,54,0,0,0,0,15,49,0,0,0,0,0,0,43,60,0,0,0,0,30,30,0,0,0,0,0,0,46,0,0,0,0,0,46,27],[24,59,0,0,0,0,15,49,0,0,0,0,0,0,43,60,0,0,0,0,30,30,0,0,0,0,0,0,27,0,0,0,0,0,27,46] >;
C24.4D4 in GAP, Magma, Sage, TeX
C_{24}._4D_4 % in TeX
G:=Group("C24.4D4"); // GroupNames label
G:=SmallGroup(192,696);
// by ID
G=gap.SmallGroup(192,696);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,224,253,344,254,387,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=1,c^2=a^12,b*a*b^-1=a^11,c*a*c^-1=a^-1,c*b*c^-1=a^12*b^-1>;
// generators/relations