metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C8⋊6(C4×S3), C24⋊C2⋊3C4, C24⋊11(C2×C4), C24⋊C4⋊7C2, (C2×C8).67D6, C6.54(C4×D4), C2.D8⋊12S3, C4⋊C4.172D6, Dic6⋊7(C2×C4), D12.9(C2×C4), C2.6(D8⋊S3), C6.D8.7C2, C22.92(S3×D4), Dic6⋊C4⋊7C2, Dic3⋊5D4.7C2, C12.44(C4○D4), C6.SD16⋊23C2, C3⋊4(SD16⋊C4), C6.44(C8⋊C22), C12.51(C22×C4), C2.6(Q16⋊S3), (C2×C24).145C22, (C2×C12).305C23, C4.12(Q8⋊3S3), (C2×Dic3).170D4, (C2×D12).85C22, C6.73(C8.C22), C2.14(Dic3⋊5D4), (C4×Dic3).37C22, (C2×Dic6).93C22, C4.45(S3×C2×C4), (C3×C2.D8)⋊9C2, (C2×C24⋊C2).7C2, (C2×C6).310(C2×D4), (C2×C3⋊C8).74C22, (C3×C4⋊C4).98C22, (C2×C4).408(C22×S3), SmallGroup(192,448)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C24⋊C2⋊C4
G = < a,b,c | a24=b2=c4=1, bab=a11, cac-1=a7, bc=cb >
Subgroups: 352 in 120 conjugacy classes, 49 normal (37 characteristic)
C1, C2, C2, C3, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, Q8, C23, Dic3, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C4⋊C4, C2×C8, C2×C8, SD16, C22×C4, C2×D4, C2×Q8, C3⋊C8, C24, Dic6, Dic6, C4×S3, D12, D12, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C8⋊C4, D4⋊C4, Q8⋊C4, C2.D8, C4×D4, C4×Q8, C2×SD16, C24⋊C2, C2×C3⋊C8, C4×Dic3, C4×Dic3, Dic3⋊C4, D6⋊C4, C3×C4⋊C4, C2×C24, C2×Dic6, S3×C2×C4, C2×D12, SD16⋊C4, C6.D8, C6.SD16, C24⋊C4, C3×C2.D8, Dic6⋊C4, Dic3⋊5D4, C2×C24⋊C2, C24⋊C2⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22×C4, C2×D4, C4○D4, C4×S3, C22×S3, C4×D4, C8⋊C22, C8.C22, S3×C2×C4, S3×D4, Q8⋊3S3, SD16⋊C4, Dic3⋊5D4, D8⋊S3, Q16⋊S3, C24⋊C2⋊C4
►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 66)(2 53)(3 64)(4 51)(5 62)(6 49)(7 60)(8 71)(9 58)(10 69)(11 56)(12 67)(13 54)(14 65)(15 52)(16 63)(17 50)(18 61)(19 72)(20 59)(21 70)(22 57)(23 68)(24 55)(25 75)(26 86)(27 73)(28 84)(29 95)(30 82)(31 93)(32 80)(33 91)(34 78)(35 89)(36 76)(37 87)(38 74)(39 85)(40 96)(41 83)(42 94)(43 81)(44 92)(45 79)(46 90)(47 77)(48 88)
(1 77 66 47)(2 84 67 30)(3 91 68 37)(4 74 69 44)(5 81 70 27)(6 88 71 34)(7 95 72 41)(8 78 49 48)(9 85 50 31)(10 92 51 38)(11 75 52 45)(12 82 53 28)(13 89 54 35)(14 96 55 42)(15 79 56 25)(16 86 57 32)(17 93 58 39)(18 76 59 46)(19 83 60 29)(20 90 61 36)(21 73 62 43)(22 80 63 26)(23 87 64 33)(24 94 65 40)
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), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,75)(26,86)(27,73)(28,84)(29,95)(30,82)(31,93)(32,80)(33,91)(34,78)(35,89)(36,76)(37,87)(38,74)(39,85)(40,96)(41,83)(42,94)(43,81)(44,92)(45,79)(46,90)(47,77)(48,88), (1,77,66,47)(2,84,67,30)(3,91,68,37)(4,74,69,44)(5,81,70,27)(6,88,71,34)(7,95,72,41)(8,78,49,48)(9,85,50,31)(10,92,51,38)(11,75,52,45)(12,82,53,28)(13,89,54,35)(14,96,55,42)(15,79,56,25)(16,86,57,32)(17,93,58,39)(18,76,59,46)(19,83,60,29)(20,90,61,36)(21,73,62,43)(22,80,63,26)(23,87,64,33)(24,94,65,40)>;
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), (1,66)(2,53)(3,64)(4,51)(5,62)(6,49)(7,60)(8,71)(9,58)(10,69)(11,56)(12,67)(13,54)(14,65)(15,52)(16,63)(17,50)(18,61)(19,72)(20,59)(21,70)(22,57)(23,68)(24,55)(25,75)(26,86)(27,73)(28,84)(29,95)(30,82)(31,93)(32,80)(33,91)(34,78)(35,89)(36,76)(37,87)(38,74)(39,85)(40,96)(41,83)(42,94)(43,81)(44,92)(45,79)(46,90)(47,77)(48,88), (1,77,66,47)(2,84,67,30)(3,91,68,37)(4,74,69,44)(5,81,70,27)(6,88,71,34)(7,95,72,41)(8,78,49,48)(9,85,50,31)(10,92,51,38)(11,75,52,45)(12,82,53,28)(13,89,54,35)(14,96,55,42)(15,79,56,25)(16,86,57,32)(17,93,58,39)(18,76,59,46)(19,83,60,29)(20,90,61,36)(21,73,62,43)(22,80,63,26)(23,87,64,33)(24,94,65,40) );
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)], [(1,66),(2,53),(3,64),(4,51),(5,62),(6,49),(7,60),(8,71),(9,58),(10,69),(11,56),(12,67),(13,54),(14,65),(15,52),(16,63),(17,50),(18,61),(19,72),(20,59),(21,70),(22,57),(23,68),(24,55),(25,75),(26,86),(27,73),(28,84),(29,95),(30,82),(31,93),(32,80),(33,91),(34,78),(35,89),(36,76),(37,87),(38,74),(39,85),(40,96),(41,83),(42,94),(43,81),(44,92),(45,79),(46,90),(47,77),(48,88)], [(1,77,66,47),(2,84,67,30),(3,91,68,37),(4,74,69,44),(5,81,70,27),(6,88,71,34),(7,95,72,41),(8,78,49,48),(9,85,50,31),(10,92,51,38),(11,75,52,45),(12,82,53,28),(13,89,54,35),(14,96,55,42),(15,79,56,25),(16,86,57,32),(17,93,58,39),(18,76,59,46),(19,83,60,29),(20,90,61,36),(21,73,62,43),(22,80,63,26),(23,87,64,33),(24,94,65,40)]])
36 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 24A | 24B | 24C | 24D |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 4 | 4 | 12 | 12 | 4 | 4 | 8 | 8 | 8 | 8 | 4 | 4 | 4 | 4 |
36 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | C4○D4 | C4×S3 | C8⋊C22 | C8.C22 | Q8⋊3S3 | S3×D4 | D8⋊S3 | Q16⋊S3 |
| kernel | C24⋊C2⋊C4 | C6.D8 | C6.SD16 | C24⋊C4 | C3×C2.D8 | Dic6⋊C4 | Dic3⋊5D4 | C2×C24⋊C2 | C24⋊C2 | C2.D8 | C2×Dic3 | C4⋊C4 | C2×C8 | C12 | C8 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 2 | 2 | 1 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 |
Matrix representation of C24⋊C2⋊C4 ►in GL6(𝔽73)
| 0 | 72 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 11 | 42 | 11 | 42 |
| 0 | 0 | 31 | 42 | 31 | 42 |
| 0 | 0 | 62 | 31 | 11 | 42 |
| 0 | 0 | 42 | 31 | 31 | 42 |
| 72 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 1 | 1 |
| 46 | 0 | 0 | 0 | 0 | 0 |
| 0 | 46 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 |
G:=sub<GL(6,GF(73))| [0,1,0,0,0,0,72,1,0,0,0,0,0,0,11,31,62,42,0,0,42,42,31,31,0,0,11,31,11,31,0,0,42,42,42,42],[72,1,0,0,0,0,0,1,0,0,0,0,0,0,1,72,0,0,0,0,0,72,0,0,0,0,0,0,72,1,0,0,0,0,0,1],[46,0,0,0,0,0,0,46,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,72,0,0,0,0,0,0,72] >;
C24⋊C2⋊C4 in GAP, Magma, Sage, TeX
C_{24}\rtimes C_2\rtimes C_4 % in TeX
G:=Group("C24:C2:C4"); // GroupNames label
G:=SmallGroup(192,448);
// by ID
G=gap.SmallGroup(192,448);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,120,219,58,1684,438,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^24=b^2=c^4=1,b*a*b=a^11,c*a*c^-1=a^7,b*c=c*b>;
// generators/relations