metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C24⋊3D4, C23.23D12, C8⋊2(C3⋊D4), C3⋊4(C8⋊2D4), C8⋊Dic3⋊4C2, (C2×C8).78D6, (C2×D24)⋊12C2, (C2×C4).52D12, C12⋊7D4⋊41C2, C12.421(C2×D4), C2.D24⋊42C2, (C2×C12).298D4, (C2×M4(2))⋊2S3, (C6×M4(2))⋊2C2, C6.74(C4⋊D4), C2.23(C8⋊D6), C6.23(C8⋊C22), (C2×C24).64C22, (C22×C6).104D4, (C22×C4).159D6, C12.231(C4○D4), C4.115(C4○D12), C2.22(C12⋊7D4), (C2×C12).776C23, (C2×D12).21C22, C22.135(C2×D12), C4⋊Dic3.27C22, (C22×C12).305C22, (C2×C6).166(C2×D4), C4.114(C2×C3⋊D4), (C2×C4).725(C22×S3), SmallGroup(192,694)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊3D4
G = < a,b,c | a24=b4=c2=1, bab-1=a11, cac=a-1, cbc=b-1 >
Subgroups: 488 in 130 conjugacy classes, 43 normal (27 characteristic)
C1, C2, C2, C2, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, D4, C23, C23, Dic3, C12, C12, D6, C2×C6, C2×C6, C22⋊C4, C4⋊C4, C2×C8, M4(2), D8, C22×C4, C2×D4, C24, C24, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C22×S3, C22×C6, D4⋊C4, C4.Q8, C4⋊D4, C2×M4(2), C2×D8, D24, C4⋊Dic3, D6⋊C4, C2×C24, C3×M4(2), C2×D12, C2×C3⋊D4, C22×C12, C8⋊2D4, C8⋊Dic3, C2.D24, C2×D24, C12⋊7D4, C6×M4(2), C24⋊3D4
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⋊2D4, C8⋊D6, C12⋊7D4, C24⋊3D4
►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 85 56 38)(2 96 57 25)(3 83 58 36)(4 94 59 47)(5 81 60 34)(6 92 61 45)(7 79 62 32)(8 90 63 43)(9 77 64 30)(10 88 65 41)(11 75 66 28)(12 86 67 39)(13 73 68 26)(14 84 69 37)(15 95 70 48)(16 82 71 35)(17 93 72 46)(18 80 49 33)(19 91 50 44)(20 78 51 31)(21 89 52 42)(22 76 53 29)(23 87 54 40)(24 74 55 27)
(2 24)(3 23)(4 22)(5 21)(6 20)(7 19)(8 18)(9 17)(10 16)(11 15)(12 14)(25 74)(26 73)(27 96)(28 95)(29 94)(30 93)(31 92)(32 91)(33 90)(34 89)(35 88)(36 87)(37 86)(38 85)(39 84)(40 83)(41 82)(42 81)(43 80)(44 79)(45 78)(46 77)(47 76)(48 75)(49 63)(50 62)(51 61)(52 60)(53 59)(54 58)(55 57)(64 72)(65 71)(66 70)(67 69)
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), (1,85,56,38)(2,96,57,25)(3,83,58,36)(4,94,59,47)(5,81,60,34)(6,92,61,45)(7,79,62,32)(8,90,63,43)(9,77,64,30)(10,88,65,41)(11,75,66,28)(12,86,67,39)(13,73,68,26)(14,84,69,37)(15,95,70,48)(16,82,71,35)(17,93,72,46)(18,80,49,33)(19,91,50,44)(20,78,51,31)(21,89,52,42)(22,76,53,29)(23,87,54,40)(24,74,55,27), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,74)(26,73)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69)>;
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), (1,85,56,38)(2,96,57,25)(3,83,58,36)(4,94,59,47)(5,81,60,34)(6,92,61,45)(7,79,62,32)(8,90,63,43)(9,77,64,30)(10,88,65,41)(11,75,66,28)(12,86,67,39)(13,73,68,26)(14,84,69,37)(15,95,70,48)(16,82,71,35)(17,93,72,46)(18,80,49,33)(19,91,50,44)(20,78,51,31)(21,89,52,42)(22,76,53,29)(23,87,54,40)(24,74,55,27), (2,24)(3,23)(4,22)(5,21)(6,20)(7,19)(8,18)(9,17)(10,16)(11,15)(12,14)(25,74)(26,73)(27,96)(28,95)(29,94)(30,93)(31,92)(32,91)(33,90)(34,89)(35,88)(36,87)(37,86)(38,85)(39,84)(40,83)(41,82)(42,81)(43,80)(44,79)(45,78)(46,77)(47,76)(48,75)(49,63)(50,62)(51,61)(52,60)(53,59)(54,58)(55,57)(64,72)(65,71)(66,70)(67,69) );
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)], [(1,85,56,38),(2,96,57,25),(3,83,58,36),(4,94,59,47),(5,81,60,34),(6,92,61,45),(7,79,62,32),(8,90,63,43),(9,77,64,30),(10,88,65,41),(11,75,66,28),(12,86,67,39),(13,73,68,26),(14,84,69,37),(15,95,70,48),(16,82,71,35),(17,93,72,46),(18,80,49,33),(19,91,50,44),(20,78,51,31),(21,89,52,42),(22,76,53,29),(23,87,54,40),(24,74,55,27)], [(2,24),(3,23),(4,22),(5,21),(6,20),(7,19),(8,18),(9,17),(10,16),(11,15),(12,14),(25,74),(26,73),(27,96),(28,95),(29,94),(30,93),(31,92),(32,91),(33,90),(34,89),(35,88),(36,87),(37,86),(38,85),(39,84),(40,83),(41,82),(42,81),(43,80),(44,79),(45,78),(46,77),(47,76),(48,75),(49,63),(50,62),(51,61),(52,60),(53,59),(54,58),(55,57),(64,72),(65,71),(66,70),(67,69)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 3 | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | ··· | 24H |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 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 | 24 | 24 | 2 | 2 | 2 | 4 | 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⋊3D4 | C8⋊Dic3 | C2.D24 | C2×D24 | C12⋊7D4 | 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 | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 1 | 1 | 2 | 1 | 2 | 4 | 2 | 2 | 4 | 2 | 4 |
Matrix representation of C24⋊3D4 ►in GL6(𝔽73)
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 16 | 3 | 49 | 32 |
| 0 | 0 | 70 | 13 | 41 | 17 |
| 0 | 0 | 16 | 3 | 57 | 70 |
| 0 | 0 | 70 | 13 | 3 | 60 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 71 | 0 |
| 0 | 0 | 72 | 72 | 2 | 2 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 72 | 72 |
G:=sub<GL(6,GF(73))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,16,70,16,70,0,0,3,13,3,13,0,0,49,41,57,3,0,0,32,17,70,60],[0,1,0,0,0,0,72,0,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,71,2,72,1,0,0,0,2,0,1],[1,0,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,1,72,0,0,0,0,0,72] >;
C24⋊3D4 in GAP, Magma, Sage, TeX
C_{24}\rtimes_3D_4 % in TeX
G:=Group("C24:3D4"); // GroupNames label
G:=SmallGroup(192,694);
// by ID
G=gap.SmallGroup(192,694);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,254,387,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^4=c^2=1,b*a*b^-1=a^11,c*a*c=a^-1,c*b*c=b^-1>;
// generators/relations