direct product, metabelian, supersoluble, monomial, 2-hyperelementary
Aliases: C4×D24, C12⋊5D8, C42.259D6, C3⋊1(C4×D8), (C4×C8)⋊7S3, C8⋊10(C4×S3), C6.2(C2×D8), C6.7(C4×D4), (C4×C24)⋊12C2, C24⋊22(C2×C4), (C4×D12)⋊1C2, D12⋊8(C2×C4), C2.1(C2×D24), C24⋊1C4⋊28C2, C6.3(C4○D8), C2.10(C4×D12), (C2×C8).286D6, (C2×C4).61D12, (C2×D24).14C2, C2.2(C4○D24), (C2×C12).351D4, C2.D24⋊43C2, C22.28(C2×D12), C12.217(C4○D4), C4.101(C4○D12), C12.102(C22×C4), (C2×C24).346C22, (C2×C12).721C23, (C4×C12).326C22, (C2×D12).188C22, C4⋊Dic3.263C22, C4.60(S3×C2×C4), (C2×C6).104(C2×D4), (C2×C4).664(C22×S3), SmallGroup(192,251)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4×D24
G = < a,b,c | a4=b24=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 456 in 134 conjugacy classes, 55 normal (31 characteristic)
C1, C2, C2, C3, C4, C4, C4, C22, C22, S3, C6, C8, C8, C2×C4, C2×C4, D4, C23, Dic3, C12, C12, C12, D6, C2×C6, C42, C22⋊C4, C4⋊C4, C2×C8, D8, C22×C4, C2×D4, C24, C24, C4×S3, D12, D12, C2×Dic3, C2×C12, C22×S3, C4×C8, D4⋊C4, C2.D8, C4×D4, C2×D8, D24, C4⋊Dic3, D6⋊C4, C4×C12, C2×C24, S3×C2×C4, C2×D12, C4×D8, C24⋊1C4, C2.D24, C4×C24, C4×D12, C2×D24, C4×D24
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, D8, C22×C4, C2×D4, C4○D4, C4×S3, D12, C22×S3, C4×D4, C2×D8, C4○D8, D24, S3×C2×C4, C2×D12, C4○D12, C4×D8, C4×D12, C2×D24, C4○D24, C4×D24
►On 96 points
(1 87 64 26)(2 88 65 27)(3 89 66 28)(4 90 67 29)(5 91 68 30)(6 92 69 31)(7 93 70 32)(8 94 71 33)(9 95 72 34)(10 96 49 35)(11 73 50 36)(12 74 51 37)(13 75 52 38)(14 76 53 39)(15 77 54 40)(16 78 55 41)(17 79 56 42)(18 80 57 43)(19 81 58 44)(20 82 59 45)(21 83 60 46)(22 84 61 47)(23 85 62 48)(24 86 63 25)
(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 54)(2 53)(3 52)(4 51)(5 50)(6 49)(7 72)(8 71)(9 70)(10 69)(11 68)(12 67)(13 66)(14 65)(15 64)(16 63)(17 62)(18 61)(19 60)(20 59)(21 58)(22 57)(23 56)(24 55)(25 78)(26 77)(27 76)(28 75)(29 74)(30 73)(31 96)(32 95)(33 94)(34 93)(35 92)(36 91)(37 90)(38 89)(39 88)(40 87)(41 86)(42 85)(43 84)(44 83)(45 82)(46 81)(47 80)(48 79)
G:=sub<Sym(96)| (1,87,64,26)(2,88,65,27)(3,89,66,28)(4,90,67,29)(5,91,68,30)(6,92,69,31)(7,93,70,32)(8,94,71,33)(9,95,72,34)(10,96,49,35)(11,73,50,36)(12,74,51,37)(13,75,52,38)(14,76,53,39)(15,77,54,40)(16,78,55,41)(17,79,56,42)(18,80,57,43)(19,81,58,44)(20,82,59,45)(21,83,60,46)(22,84,61,47)(23,85,62,48)(24,86,63,25), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)>;
G:=Group( (1,87,64,26)(2,88,65,27)(3,89,66,28)(4,90,67,29)(5,91,68,30)(6,92,69,31)(7,93,70,32)(8,94,71,33)(9,95,72,34)(10,96,49,35)(11,73,50,36)(12,74,51,37)(13,75,52,38)(14,76,53,39)(15,77,54,40)(16,78,55,41)(17,79,56,42)(18,80,57,43)(19,81,58,44)(20,82,59,45)(21,83,60,46)(22,84,61,47)(23,85,62,48)(24,86,63,25), (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), (1,54)(2,53)(3,52)(4,51)(5,50)(6,49)(7,72)(8,71)(9,70)(10,69)(11,68)(12,67)(13,66)(14,65)(15,64)(16,63)(17,62)(18,61)(19,60)(20,59)(21,58)(22,57)(23,56)(24,55)(25,78)(26,77)(27,76)(28,75)(29,74)(30,73)(31,96)(32,95)(33,94)(34,93)(35,92)(36,91)(37,90)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79) );
G=PermutationGroup([[(1,87,64,26),(2,88,65,27),(3,89,66,28),(4,90,67,29),(5,91,68,30),(6,92,69,31),(7,93,70,32),(8,94,71,33),(9,95,72,34),(10,96,49,35),(11,73,50,36),(12,74,51,37),(13,75,52,38),(14,76,53,39),(15,77,54,40),(16,78,55,41),(17,79,56,42),(18,80,57,43),(19,81,58,44),(20,82,59,45),(21,83,60,46),(22,84,61,47),(23,85,62,48),(24,86,63,25)], [(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)], [(1,54),(2,53),(3,52),(4,51),(5,50),(6,49),(7,72),(8,71),(9,70),(10,69),(11,68),(12,67),(13,66),(14,65),(15,64),(16,63),(17,62),(18,61),(19,60),(20,59),(21,58),(22,57),(23,56),(24,55),(25,78),(26,77),(27,76),(28,75),(29,74),(30,73),(31,96),(32,95),(33,94),(34,93),(35,92),(36,91),(37,90),(38,89),(39,88),(40,87),(41,86),(42,85),(43,84),(44,83),(45,82),(46,81),(47,80),(48,79)]])
60 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 4K | 4L | 6A | 6B | 6C | 8A | ··· | 8H | 12A | ··· | 12L | 24A | ··· | 24P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 8 | ··· | 8 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 12 | 12 | 2 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
60 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | ||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | D4 | D6 | D6 | D8 | C4○D4 | C4×S3 | D12 | C4○D8 | D24 | C4○D12 | C4○D24 |
| kernel | C4×D24 | C24⋊1C4 | C2.D24 | C4×C24 | C4×D12 | C2×D24 | D24 | C4×C8 | C2×C12 | C42 | C2×C8 | C12 | C12 | C8 | C2×C4 | C6 | C4 | C4 | C2 |
| # reps | 1 | 1 | 2 | 1 | 2 | 1 | 8 | 1 | 2 | 1 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 4 | 8 |
Matrix representation of C4×D24 ►in GL3(𝔽73) generated by
| 46 | 0 | 0 |
| 0 | 46 | 0 |
| 0 | 0 | 46 |
| 1 | 0 | 0 |
| 0 | 68 | 50 |
| 0 | 23 | 18 |
| 1 | 0 | 0 |
| 0 | 14 | 7 |
| 0 | 66 | 59 |
G:=sub<GL(3,GF(73))| [46,0,0,0,46,0,0,0,46],[1,0,0,0,68,23,0,50,18],[1,0,0,0,14,66,0,7,59] >;
C4×D24 in GAP, Magma, Sage, TeX
C_4\times D_{24} % in TeX
G:=Group("C4xD24"); // GroupNames label
G:=SmallGroup(192,251);
// by ID
G=gap.SmallGroup(192,251);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-3,253,344,58,1684,102,6278]);
// Polycyclic
G:=Group<a,b,c|a^4=b^24=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations