direct product, metabelian, supersoluble, monomial
Aliases: C3×D24⋊C2, D24⋊5C6, C24.58D6, (S3×C8)⋊3C6, (S3×C24)⋊7C2, C8.10(S3×C6), C24.8(C2×C6), (C3×Q16)⋊7S3, (C3×Q16)⋊3C6, Q16⋊3(C3×S3), D6.3(C3×D4), C6.36(C6×D4), (C3×D24)⋊13C2, Q8⋊2S3⋊4C6, Q8⋊3S3⋊6C6, D12.5(C2×C6), (S3×C6).27D4, C6.196(S3×D4), Q8.15(S3×C6), (C3×Q8).52D6, C32⋊22(C4○D8), (C32×Q16)⋊4C2, (C3×C24).28C22, C12.10(C22×C6), (C3×C12).81C23, (C3×Dic3).51D4, Dic3.14(C3×D4), (S3×C12).52C22, C12.161(C22×S3), (C3×D12).29C22, (Q8×C32).15C22, C3⋊4(C3×C4○D8), C3⋊C8.8(C2×C6), C4.10(S3×C2×C6), C2.24(C3×S3×D4), (C4×S3).12(C2×C6), (C3×Q8⋊3S3)⋊6C2, (C3×C6).224(C2×D4), (C3×C3⋊C8).40C22, (C3×Q8).10(C2×C6), (C3×Q8⋊2S3)⋊14C2, SmallGroup(288,690)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D24⋊C2
G = < a,b,c,d | a3=b24=c2=d2=1, ab=ba, ac=ca, ad=da, cbc=b-1, dbd=b17, dcd=b4c >
Subgroups: 346 in 133 conjugacy classes, 54 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, D4, Q8, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C8, D8, SD16, Q16, C4○D4, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C4×S3, D12, D12, C2×C12, C3×D4, C3×Q8, C3×Q8, C4○D8, C3×Dic3, C3×C12, C3×C12, S3×C6, S3×C6, S3×C8, D24, Q8⋊2S3, C2×C24, C3×D8, C3×SD16, C3×Q16, C3×Q16, Q8⋊3S3, C3×C4○D4, C3×C3⋊C8, C3×C24, S3×C12, S3×C12, C3×D12, C3×D12, Q8×C32, D24⋊C2, C3×C4○D8, S3×C24, C3×D24, C3×Q8⋊2S3, C32×Q16, C3×Q8⋊3S3, C3×D24⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, C3×D4, C22×S3, C22×C6, C4○D8, S3×C6, S3×D4, C6×D4, S3×C2×C6, D24⋊C2, C3×C4○D8, C3×S3×D4, C3×D24⋊C2
►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 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)(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 81 89)(74 82 90)(75 83 91)(76 84 92)(77 85 93)(78 86 94)(79 87 95)(80 88 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 52)(2 51)(3 50)(4 49)(5 72)(6 71)(7 70)(8 69)(9 68)(10 67)(11 66)(12 65)(13 64)(14 63)(15 62)(16 61)(17 60)(18 59)(19 58)(20 57)(21 56)(22 55)(23 54)(24 53)(25 85)(26 84)(27 83)(28 82)(29 81)(30 80)(31 79)(32 78)(33 77)(34 76)(35 75)(36 74)(37 73)(38 96)(39 95)(40 94)(41 93)(42 92)(43 91)(44 90)(45 89)(46 88)(47 87)(48 86)
(1 90)(2 83)(3 76)(4 93)(5 86)(6 79)(7 96)(8 89)(9 82)(10 75)(11 92)(12 85)(13 78)(14 95)(15 88)(16 81)(17 74)(18 91)(19 84)(20 77)(21 94)(22 87)(23 80)(24 73)(25 61)(26 54)(27 71)(28 64)(29 57)(30 50)(31 67)(32 60)(33 53)(34 70)(35 63)(36 56)(37 49)(38 66)(39 59)(40 52)(41 69)(42 62)(43 55)(44 72)(45 65)(46 58)(47 51)(48 68)
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)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96), (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), (1,52)(2,51)(3,50)(4,49)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86), (1,90)(2,83)(3,76)(4,93)(5,86)(6,79)(7,96)(8,89)(9,82)(10,75)(11,92)(12,85)(13,78)(14,95)(15,88)(16,81)(17,74)(18,91)(19,84)(20,77)(21,94)(22,87)(23,80)(24,73)(25,61)(26,54)(27,71)(28,64)(29,57)(30,50)(31,67)(32,60)(33,53)(34,70)(35,63)(36,56)(37,49)(38,66)(39,59)(40,52)(41,69)(42,62)(43,55)(44,72)(45,65)(46,58)(47,51)(48,68)>;
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)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40)(49,57,65)(50,58,66)(51,59,67)(52,60,68)(53,61,69)(54,62,70)(55,63,71)(56,64,72)(73,81,89)(74,82,90)(75,83,91)(76,84,92)(77,85,93)(78,86,94)(79,87,95)(80,88,96), (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), (1,52)(2,51)(3,50)(4,49)(5,72)(6,71)(7,70)(8,69)(9,68)(10,67)(11,66)(12,65)(13,64)(14,63)(15,62)(16,61)(17,60)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,85)(26,84)(27,83)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,76)(35,75)(36,74)(37,73)(38,96)(39,95)(40,94)(41,93)(42,92)(43,91)(44,90)(45,89)(46,88)(47,87)(48,86), (1,90)(2,83)(3,76)(4,93)(5,86)(6,79)(7,96)(8,89)(9,82)(10,75)(11,92)(12,85)(13,78)(14,95)(15,88)(16,81)(17,74)(18,91)(19,84)(20,77)(21,94)(22,87)(23,80)(24,73)(25,61)(26,54)(27,71)(28,64)(29,57)(30,50)(31,67)(32,60)(33,53)(34,70)(35,63)(36,56)(37,49)(38,66)(39,59)(40,52)(41,69)(42,62)(43,55)(44,72)(45,65)(46,58)(47,51)(48,68) );
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),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40),(49,57,65),(50,58,66),(51,59,67),(52,60,68),(53,61,69),(54,62,70),(55,63,71),(56,64,72),(73,81,89),(74,82,90),(75,83,91),(76,84,92),(77,85,93),(78,86,94),(79,87,95),(80,88,96)], [(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)], [(1,52),(2,51),(3,50),(4,49),(5,72),(6,71),(7,70),(8,69),(9,68),(10,67),(11,66),(12,65),(13,64),(14,63),(15,62),(16,61),(17,60),(18,59),(19,58),(20,57),(21,56),(22,55),(23,54),(24,53),(25,85),(26,84),(27,83),(28,82),(29,81),(30,80),(31,79),(32,78),(33,77),(34,76),(35,75),(36,74),(37,73),(38,96),(39,95),(40,94),(41,93),(42,92),(43,91),(44,90),(45,89),(46,88),(47,87),(48,86)], [(1,90),(2,83),(3,76),(4,93),(5,86),(6,79),(7,96),(8,89),(9,82),(10,75),(11,92),(12,85),(13,78),(14,95),(15,88),(16,81),(17,74),(18,91),(19,84),(20,77),(21,94),(22,87),(23,80),(24,73),(25,61),(26,54),(27,71),(28,64),(29,57),(30,50),(31,67),(32,60),(33,53),(34,70),(35,63),(36,56),(37,49),(38,66),(39,59),(40,52),(41,69),(42,62),(43,55),(44,72),(45,65),(46,58),(47,51),(48,68)]])
63 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 6J | 6K | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 12E | 12F | 12G | ··· | 12M | 12N | ··· | 12S | 24A | 24B | 24C | 24D | 24E | ··· | 24J | 24K | 24L | 24M | 24N |
| order | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | 24 | 24 | 24 | 24 | ··· | 24 | 24 | 24 | 24 | 24 |
| size | 1 | 1 | 6 | 12 | 12 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 4 | 4 | 1 | 1 | 2 | 2 | 2 | 6 | 6 | 12 | 12 | 12 | 12 | 2 | 2 | 6 | 6 | 2 | 2 | 3 | 3 | 3 | 3 | 4 | ··· | 4 | 8 | ··· | 8 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
63 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3×D4 | C3×D4 | C4○D8 | S3×C6 | S3×C6 | C3×C4○D8 | S3×D4 | D24⋊C2 | C3×S3×D4 | C3×D24⋊C2 |
| kernel | C3×D24⋊C2 | S3×C24 | C3×D24 | C3×Q8⋊2S3 | C32×Q16 | C3×Q8⋊3S3 | D24⋊C2 | S3×C8 | D24 | Q8⋊2S3 | C3×Q16 | Q8⋊3S3 | C3×Q16 | C3×Dic3 | S3×C6 | C24 | C3×Q8 | Q16 | Dic3 | D6 | C32 | C8 | Q8 | C3 | C6 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 8 | 1 | 2 | 2 | 4 |
Matrix representation of C3×D24⋊C2 ►in GL4(𝔽7) generated by
| 2 | 0 | 0 | 0 |
| 0 | 2 | 0 | 0 |
| 0 | 0 | 2 | 0 |
| 0 | 0 | 0 | 2 |
| 0 | 3 | 0 | 6 |
| 5 | 6 | 6 | 1 |
| 2 | 2 | 1 | 3 |
| 4 | 6 | 6 | 4 |
| 4 | 2 | 4 | 6 |
| 4 | 5 | 1 | 2 |
| 1 | 1 | 3 | 4 |
| 6 | 1 | 2 | 2 |
| 3 | 1 | 5 | 6 |
| 4 | 3 | 6 | 4 |
| 1 | 4 | 5 | 6 |
| 3 | 5 | 4 | 3 |
G:=sub<GL(4,GF(7))| [2,0,0,0,0,2,0,0,0,0,2,0,0,0,0,2],[0,5,2,4,3,6,2,6,0,6,1,6,6,1,3,4],[4,4,1,6,2,5,1,1,4,1,3,2,6,2,4,2],[3,4,1,3,1,3,4,5,5,6,5,4,6,4,6,3] >;
C3×D24⋊C2 in GAP, Magma, Sage, TeX
C_3\times D_{24}\rtimes C_2 % in TeX
G:=Group("C3xD24:C2"); // GroupNames label
G:=SmallGroup(288,690);
// by ID
G=gap.SmallGroup(288,690);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,701,344,1094,303,268,1271,648,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^24=c^2=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,c*b*c=b^-1,d*b*d=b^17,d*c*d=b^4*c>;
// generators/relations