direct product, metacyclic, supersoluble, monomial
Aliases: C3×C24⋊1C4, C24⋊1C12, C24⋊3Dic3, C6.22D24, C62.79D4, C6.10Dic12, C12.32Dic6, (C3×C24)⋊6C4, C6.4(C3×D8), (C2×C24).9C6, C8⋊1(C3×Dic3), (C3×C6).21D8, C2.1(C3×D24), C6.2(C3×Q16), (C3×C6).9Q16, C12.5(C3×Q8), (C2×C24).20S3, (C6×C24).11C2, (C2×C6).70D12, C4⋊Dic3.3C6, (C3×C12).23Q8, C4.7(C6×Dic3), C4.5(C3×Dic6), C12.42(C2×C12), C32⋊8(C2.D8), (C2×C12).433D6, C2.2(C3×Dic12), C22.9(C3×D12), C6.19(C4⋊Dic3), C12.64(C2×Dic3), (C6×C12).317C22, C6.6(C3×C4⋊C4), C3⋊2(C3×C2.D8), (C2×C8).3(C3×S3), (C2×C4).72(S3×C6), (C2×C6).18(C3×D4), C2.4(C3×C4⋊Dic3), (C3×C6).32(C4⋊C4), (C3×C12).128(C2×C4), (C2×C12).102(C2×C6), (C3×C4⋊Dic3).22C2, SmallGroup(288,252)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C24⋊1C4
G = < a,b,c | a3=b24=c4=1, ab=ba, ac=ca, cbc-1=b-1 >
Subgroups: 186 in 83 conjugacy classes, 50 normal (38 characteristic)
C1, C2, C3, C3, C4, C4, C22, C6, C6, C8, C2×C4, C2×C4, C32, Dic3, C12, C12, C2×C6, C2×C6, C4⋊C4, C2×C8, C3×C6, C24, C24, C2×Dic3, C2×C12, C2×C12, C2.D8, C3×Dic3, C3×C12, C62, C4⋊Dic3, C3×C4⋊C4, C2×C24, C2×C24, C3×C24, C6×Dic3, C6×C12, C24⋊1C4, C3×C2.D8, C3×C4⋊Dic3, C6×C24, C3×C24⋊1C4
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, D4, Q8, Dic3, C12, D6, C2×C6, C4⋊C4, D8, Q16, C3×S3, Dic6, D12, C2×Dic3, C2×C12, C3×D4, C3×Q8, C2.D8, C3×Dic3, S3×C6, D24, Dic12, C4⋊Dic3, C3×C4⋊C4, C3×D8, C3×Q16, C3×Dic6, C3×D12, C6×Dic3, C24⋊1C4, C3×C2.D8, C3×D24, C3×Dic12, C3×C4⋊Dic3, C3×C24⋊1C4
►On 96 points
(1 9 17)(2 10 18)(3 11 19)(4 12 20)(5 13 21)(6 14 22)(7 15 23)(8 16 24)(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 65 57)(50 66 58)(51 67 59)(52 68 60)(53 69 61)(54 70 62)(55 71 63)(56 72 64)(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 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 82 30 54)(2 81 31 53)(3 80 32 52)(4 79 33 51)(5 78 34 50)(6 77 35 49)(7 76 36 72)(8 75 37 71)(9 74 38 70)(10 73 39 69)(11 96 40 68)(12 95 41 67)(13 94 42 66)(14 93 43 65)(15 92 44 64)(16 91 45 63)(17 90 46 62)(18 89 47 61)(19 88 48 60)(20 87 25 59)(21 86 26 58)(22 85 27 57)(23 84 28 56)(24 83 29 55)
G:=sub<Sym(96)| (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,82,30,54)(2,81,31,53)(3,80,32,52)(4,79,33,51)(5,78,34,50)(6,77,35,49)(7,76,36,72)(8,75,37,71)(9,74,38,70)(10,73,39,69)(11,96,40,68)(12,95,41,67)(13,94,42,66)(14,93,43,65)(15,92,44,64)(16,91,45,63)(17,90,46,62)(18,89,47,61)(19,88,48,60)(20,87,25,59)(21,86,26,58)(22,85,27,57)(23,84,28,56)(24,83,29,55)>;
G:=Group( (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,33,41)(26,34,42)(27,35,43)(28,36,44)(29,37,45)(30,38,46)(31,39,47)(32,40,48)(49,65,57)(50,66,58)(51,67,59)(52,68,60)(53,69,61)(54,70,62)(55,71,63)(56,72,64)(73,89,81)(74,90,82)(75,91,83)(76,92,84)(77,93,85)(78,94,86)(79,95,87)(80,96,88), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24)(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), (1,82,30,54)(2,81,31,53)(3,80,32,52)(4,79,33,51)(5,78,34,50)(6,77,35,49)(7,76,36,72)(8,75,37,71)(9,74,38,70)(10,73,39,69)(11,96,40,68)(12,95,41,67)(13,94,42,66)(14,93,43,65)(15,92,44,64)(16,91,45,63)(17,90,46,62)(18,89,47,61)(19,88,48,60)(20,87,25,59)(21,86,26,58)(22,85,27,57)(23,84,28,56)(24,83,29,55) );
G=PermutationGroup([[(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,33,41),(26,34,42),(27,35,43),(28,36,44),(29,37,45),(30,38,46),(31,39,47),(32,40,48),(49,65,57),(50,66,58),(51,67,59),(52,68,60),(53,69,61),(54,70,62),(55,71,63),(56,72,64),(73,89,81),(74,90,82),(75,91,83),(76,92,84),(77,93,85),(78,94,86),(79,95,87),(80,96,88)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24),(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)], [(1,82,30,54),(2,81,31,53),(3,80,32,52),(4,79,33,51),(5,78,34,50),(6,77,35,49),(7,76,36,72),(8,75,37,71),(9,74,38,70),(10,73,39,69),(11,96,40,68),(12,95,41,67),(13,94,42,66),(14,93,43,65),(15,92,44,64),(16,91,45,63),(17,90,46,62),(18,89,47,61),(19,88,48,60),(20,87,25,59),(21,86,26,58),(22,85,27,57),(23,84,28,56),(24,83,29,55)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 6A | ··· | 6F | 6G | ··· | 6O | 8A | 8B | 8C | 8D | 12A | ··· | 12P | 12Q | ··· | 12X | 24A | ··· | 24AF |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | 8 | 8 | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 12 | 12 | 12 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 12 | ··· | 12 | 2 | ··· | 2 |
90 irreducible representations
| dim | 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 | C3 | C4 | C6 | C6 | C12 | S3 | Q8 | D4 | Dic3 | D6 | D8 | Q16 | C3×S3 | Dic6 | C3×Q8 | D12 | C3×D4 | C3×Dic3 | S3×C6 | D24 | Dic12 | C3×D8 | C3×Q16 | C3×Dic6 | C3×D12 | C3×D24 | C3×Dic12 |
| kernel | C3×C24⋊1C4 | C3×C4⋊Dic3 | C6×C24 | C24⋊1C4 | C3×C24 | C4⋊Dic3 | C2×C24 | C24 | C2×C24 | C3×C12 | C62 | C24 | C2×C12 | C3×C6 | C3×C6 | C2×C8 | C12 | C12 | C2×C6 | C2×C6 | C8 | C2×C4 | C6 | C6 | C6 | C6 | C4 | C22 | C2 | C2 |
| # reps | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 | 1 | 1 | 1 | 2 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 8 | 8 |
Matrix representation of C3×C24⋊1C4 ►in GL3(𝔽73) generated by
| 8 | 0 | 0 |
| 0 | 64 | 0 |
| 0 | 0 | 64 |
| 1 | 0 | 0 |
| 0 | 7 | 0 |
| 0 | 0 | 21 |
| 27 | 0 | 0 |
| 0 | 0 | 1 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(73))| [8,0,0,0,64,0,0,0,64],[1,0,0,0,7,0,0,0,21],[27,0,0,0,0,1,0,1,0] >;
C3×C24⋊1C4 in GAP, Magma, Sage, TeX
C_3\times C_{24}\rtimes_1C_4 % in TeX
G:=Group("C3xC24:1C4"); // GroupNames label
G:=SmallGroup(288,252);
// by ID
G=gap.SmallGroup(288,252);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,84,365,512,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^3=b^24=c^4=1,a*b=b*a,a*c=c*a,c*b*c^-1=b^-1>;
// generators/relations