direct product, metabelian, supersoluble, monomial
Aliases: C3×C2.D24, D12⋊2C12, C6.23D24, C62.80D4, (C2×C24)⋊2C6, (C6×C24)⋊2C2, (C2×C24)⋊2S3, C6.5(C3×D8), C4.8(S3×C12), C4⋊Dic3⋊1C6, C2.2(C3×D24), (C3×C6).22D8, (C3×D12)⋊11C4, C12.86(C4×S3), (C2×D12).1C6, (C2×C6).71D12, C12.54(C3×D4), C6.3(C3×SD16), C12.18(C2×C12), C6.48(D6⋊C4), (C6×D12).15C2, (C2×C12).435D6, (C3×C12).157D4, (C3×C6).17SD16, C6.15(C24⋊C2), C22.10(C3×D12), C12.137(C3⋊D4), C32⋊10(D4⋊C4), (C6×C12).318C22, (C2×C8)⋊2(C3×S3), C2.8(C3×D6⋊C4), C3⋊2(C3×D4⋊C4), C2.3(C3×C24⋊C2), (C2×C4).74(S3×C6), (C2×C6).19(C3×D4), C4.20(C3×C3⋊D4), C6.7(C3×C22⋊C4), (C3×C12).98(C2×C4), (C3×C4⋊Dic3)⋊25C2, (C2×C12).103(C2×C6), (C3×C6).47(C22⋊C4), SmallGroup(288,255)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C2.D24
G = < a,b,c,d | a3=b2=c24=1, d2=b, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd-1=bc-1 >
Subgroups: 346 in 111 conjugacy classes, 46 normal (42 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C8, C2×C4, C2×C4, D4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C4⋊C4, C2×C8, C2×D4, C3×S3, C3×C6, C24, D12, D12, C2×Dic3, C2×C12, C2×C12, C3×D4, C22×S3, C22×C6, D4⋊C4, C3×Dic3, C3×C12, S3×C6, C62, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×C24, C2×D12, C6×D4, C3×C24, C3×D12, C3×D12, C6×Dic3, C6×C12, S3×C2×C6, C2.D24, C3×D4⋊C4, C3×C4⋊Dic3, C6×C24, C6×D12, C3×C2.D24
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, C12, D6, C2×C6, C22⋊C4, D8, SD16, C3×S3, C4×S3, D12, C3⋊D4, C2×C12, C3×D4, D4⋊C4, S3×C6, C24⋊C2, D24, D6⋊C4, C3×C22⋊C4, C3×D8, C3×SD16, S3×C12, C3×D12, C3×C3⋊D4, C2.D24, C3×D4⋊C4, C3×C24⋊C2, C3×D24, C3×D6⋊C4, C3×C2.D24
►On 96 points
(1 17 9)(2 18 10)(3 19 11)(4 20 12)(5 21 13)(6 22 14)(7 23 15)(8 24 16)(25 33 41)(26 34 42)(27 35 43)(28 36 44)(29 37 45)(30 38 46)(31 39 47)(32 40 48)(49 57 65)(50 58 66)(51 59 67)(52 60 68)(53 61 69)(54 62 70)(55 63 71)(56 64 72)(73 89 81)(74 90 82)(75 91 83)(76 92 84)(77 93 85)(78 94 86)(79 95 87)(80 96 88)
(1 73)(2 74)(3 75)(4 76)(5 77)(6 78)(7 79)(8 80)(9 81)(10 82)(11 83)(12 84)(13 85)(14 86)(15 87)(16 88)(17 89)(18 90)(19 91)(20 92)(21 93)(22 94)(23 95)(24 96)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 49)(38 50)(39 51)(40 52)(41 53)(42 54)(43 55)(44 56)(45 57)(46 58)(47 59)(48 60)
(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 58 73 46)(2 45 74 57)(3 56 75 44)(4 43 76 55)(5 54 77 42)(6 41 78 53)(7 52 79 40)(8 39 80 51)(9 50 81 38)(10 37 82 49)(11 72 83 36)(12 35 84 71)(13 70 85 34)(14 33 86 69)(15 68 87 32)(16 31 88 67)(17 66 89 30)(18 29 90 65)(19 64 91 28)(20 27 92 63)(21 62 93 26)(22 25 94 61)(23 60 95 48)(24 47 96 59)
G:=sub<Sym(96)| (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,58,73,46)(2,45,74,57)(3,56,75,44)(4,43,76,55)(5,54,77,42)(6,41,78,53)(7,52,79,40)(8,39,80,51)(9,50,81,38)(10,37,82,49)(11,72,83,36)(12,35,84,71)(13,70,85,34)(14,33,86,69)(15,68,87,32)(16,31,88,67)(17,66,89,30)(18,29,90,65)(19,64,91,28)(20,27,92,63)(21,62,93,26)(22,25,94,61)(23,60,95,48)(24,47,96,59)>;
G:=Group( (1,17,9)(2,18,10)(3,19,11)(4,20,12)(5,21,13)(6,22,14)(7,23,15)(8,24,16)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,73)(2,74)(3,75)(4,76)(5,77)(6,78)(7,79)(8,80)(9,81)(10,82)(11,83)(12,84)(13,85)(14,86)(15,87)(16,88)(17,89)(18,90)(19,91)(20,92)(21,93)(22,94)(23,95)(24,96)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,49)(38,50)(39,51)(40,52)(41,53)(42,54)(43,55)(44,56)(45,57)(46,58)(47,59)(48,60), (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), (1,58,73,46)(2,45,74,57)(3,56,75,44)(4,43,76,55)(5,54,77,42)(6,41,78,53)(7,52,79,40)(8,39,80,51)(9,50,81,38)(10,37,82,49)(11,72,83,36)(12,35,84,71)(13,70,85,34)(14,33,86,69)(15,68,87,32)(16,31,88,67)(17,66,89,30)(18,29,90,65)(19,64,91,28)(20,27,92,63)(21,62,93,26)(22,25,94,61)(23,60,95,48)(24,47,96,59) );
G=PermutationGroup([[(1,17,9),(2,18,10),(3,19,11),(4,20,12),(5,21,13),(6,22,14),(7,23,15),(8,24,16),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,73),(2,74),(3,75),(4,76),(5,77),(6,78),(7,79),(8,80),(9,81),(10,82),(11,83),(12,84),(13,85),(14,86),(15,87),(16,88),(17,89),(18,90),(19,91),(20,92),(21,93),(22,94),(23,95),(24,96),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,49),(38,50),(39,51),(40,52),(41,53),(42,54),(43,55),(44,56),(45,57),(46,58),(47,59),(48,60)], [(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)], [(1,58,73,46),(2,45,74,57),(3,56,75,44),(4,43,76,55),(5,54,77,42),(6,41,78,53),(7,52,79,40),(8,39,80,51),(9,50,81,38),(10,37,82,49),(11,72,83,36),(12,35,84,71),(13,70,85,34),(14,33,86,69),(15,68,87,32),(16,31,88,67),(17,66,89,30),(18,29,90,65),(19,64,91,28),(20,27,92,63),(21,62,93,26),(22,25,94,61),(23,60,95,48),(24,47,96,59)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6O | 6P | 6Q | 6R | 6S | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | 12R | 12S | 12T | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | 12 | 12 | 12 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C6 | C6 | C6 | C12 | S3 | D4 | D4 | D6 | D8 | SD16 | C3×S3 | C4×S3 | C3⋊D4 | C3×D4 | D12 | C3×D4 | S3×C6 | C24⋊C2 | D24 | C3×D8 | C3×SD16 | S3×C12 | C3×C3⋊D4 | C3×D12 | C3×C24⋊C2 | C3×D24 |
| kernel | C3×C2.D24 | C3×C4⋊Dic3 | C6×C24 | C6×D12 | C2.D24 | C3×D12 | C4⋊Dic3 | C2×C24 | C2×D12 | D12 | C2×C24 | C3×C12 | C62 | C2×C12 | C3×C6 | C3×C6 | C2×C8 | C12 | C12 | C12 | C2×C6 | C2×C6 | C2×C4 | C6 | C6 | C6 | C6 | C4 | C4 | C22 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 2 | 8 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C2.D24 ►in GL3(𝔽73) generated by
| 64 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 72 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 1 |
| 46 | 0 | 0 |
| 0 | 21 | 51 |
| 0 | 0 | 7 |
| 27 | 0 | 0 |
| 0 | 35 | 10 |
| 0 | 9 | 38 |
G:=sub<GL(3,GF(73))| [64,0,0,0,64,0,0,0,64],[72,0,0,0,1,0,0,0,1],[46,0,0,0,21,0,0,51,7],[27,0,0,0,35,9,0,10,38] >;
C3×C2.D24 in GAP, Magma, Sage, TeX
C_3\times C_2.D_{24} % in TeX
G:=Group("C3xC2.D24"); // GroupNames label
G:=SmallGroup(288,255);
// by ID
G=gap.SmallGroup(288,255);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,260,1683,136,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^2=c^24=1,d^2=b,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=b*c^-1>;
// generators/relations