direct product, metacyclic, supersoluble, monomial
Aliases: C3×C48⋊C2, C48⋊2C6, C48⋊4S3, D24.1C6, C6.20D24, C24.80D6, C32⋊5SD32, Dic12⋊1C6, C12.63D12, C16⋊2(C3×S3), (C3×C48)⋊4C2, C6.2(C3×D8), C8.14(S3×C6), C3⋊1(C3×SD32), C2.4(C3×D24), C4.2(C3×D12), (C3×C6).19D8, C24.17(C2×C6), (C3×D24).1C2, C12.25(C3×D4), (C3×Dic12)⋊4C2, (C3×C12).127D4, (C3×C24).55C22, SmallGroup(288,234)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×C48⋊C2
G = < a,b,c | a3=b48=c2=1, ab=ba, ac=ca, cbc=b23 >
Subgroups: 214 in 57 conjugacy classes, 26 normal (all characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, D4, Q8, C32, Dic3, C12, C12, D6, C2×C6, C16, D8, Q16, C3×S3, C3×C6, C24, C24, Dic6, D12, C3×D4, C3×Q8, SD32, C3×Dic3, C3×C12, S3×C6, C48, C48, D24, Dic12, C3×D8, C3×Q16, C3×C24, C3×Dic6, C3×D12, C48⋊C2, C3×SD32, C3×C48, C3×D24, C3×Dic12, C3×C48⋊C2
Quotients: C1, C2, C3, C22, S3, C6, D4, D6, C2×C6, D8, C3×S3, D12, C3×D4, SD32, S3×C6, D24, C3×D8, C3×D12, C48⋊C2, C3×SD32, C3×D24, C3×C48⋊C2
►On 96 points
Generators in S96
(1 17 33)(2 18 34)(3 19 35)(4 20 36)(5 21 37)(6 22 38)(7 23 39)(8 24 40)(9 25 41)(10 26 42)(11 27 43)(12 28 44)(13 29 45)(14 30 46)(15 31 47)(16 32 48)(49 81 65)(50 82 66)(51 83 67)(52 84 68)(53 85 69)(54 86 70)(55 87 71)(56 88 72)(57 89 73)(58 90 74)(59 91 75)(60 92 76)(61 93 77)(62 94 78)(63 95 79)(64 96 80)
(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 60)(2 83)(3 58)(4 81)(5 56)(6 79)(7 54)(8 77)(9 52)(10 75)(11 50)(12 73)(13 96)(14 71)(15 94)(16 69)(17 92)(18 67)(19 90)(20 65)(21 88)(22 63)(23 86)(24 61)(25 84)(26 59)(27 82)(28 57)(29 80)(30 55)(31 78)(32 53)(33 76)(34 51)(35 74)(36 49)(37 72)(38 95)(39 70)(40 93)(41 68)(42 91)(43 66)(44 89)(45 64)(46 87)(47 62)(48 85)
G:=sub<Sym(96)| (1,17,33)(2,18,34)(3,19,35)(4,20,36)(5,21,37)(6,22,38)(7,23,39)(8,24,40)(9,25,41)(10,26,42)(11,27,43)(12,28,44)(13,29,45)(14,30,46)(15,31,47)(16,32,48)(49,81,65)(50,82,66)(51,83,67)(52,84,68)(53,85,69)(54,86,70)(55,87,71)(56,88,72)(57,89,73)(58,90,74)(59,91,75)(60,92,76)(61,93,77)(62,94,78)(63,95,79)(64,96,80), (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), (1,60)(2,83)(3,58)(4,81)(5,56)(6,79)(7,54)(8,77)(9,52)(10,75)(11,50)(12,73)(13,96)(14,71)(15,94)(16,69)(17,92)(18,67)(19,90)(20,65)(21,88)(22,63)(23,86)(24,61)(25,84)(26,59)(27,82)(28,57)(29,80)(30,55)(31,78)(32,53)(33,76)(34,51)(35,74)(36,49)(37,72)(38,95)(39,70)(40,93)(41,68)(42,91)(43,66)(44,89)(45,64)(46,87)(47,62)(48,85)>;
G:=Group( (1,17,33)(2,18,34)(3,19,35)(4,20,36)(5,21,37)(6,22,38)(7,23,39)(8,24,40)(9,25,41)(10,26,42)(11,27,43)(12,28,44)(13,29,45)(14,30,46)(15,31,47)(16,32,48)(49,81,65)(50,82,66)(51,83,67)(52,84,68)(53,85,69)(54,86,70)(55,87,71)(56,88,72)(57,89,73)(58,90,74)(59,91,75)(60,92,76)(61,93,77)(62,94,78)(63,95,79)(64,96,80), (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), (1,60)(2,83)(3,58)(4,81)(5,56)(6,79)(7,54)(8,77)(9,52)(10,75)(11,50)(12,73)(13,96)(14,71)(15,94)(16,69)(17,92)(18,67)(19,90)(20,65)(21,88)(22,63)(23,86)(24,61)(25,84)(26,59)(27,82)(28,57)(29,80)(30,55)(31,78)(32,53)(33,76)(34,51)(35,74)(36,49)(37,72)(38,95)(39,70)(40,93)(41,68)(42,91)(43,66)(44,89)(45,64)(46,87)(47,62)(48,85) );
G=PermutationGroup([[(1,17,33),(2,18,34),(3,19,35),(4,20,36),(5,21,37),(6,22,38),(7,23,39),(8,24,40),(9,25,41),(10,26,42),(11,27,43),(12,28,44),(13,29,45),(14,30,46),(15,31,47),(16,32,48),(49,81,65),(50,82,66),(51,83,67),(52,84,68),(53,85,69),(54,86,70),(55,87,71),(56,88,72),(57,89,73),(58,90,74),(59,91,75),(60,92,76),(61,93,77),(62,94,78),(63,95,79),(64,96,80)], [(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)], [(1,60),(2,83),(3,58),(4,81),(5,56),(6,79),(7,54),(8,77),(9,52),(10,75),(11,50),(12,73),(13,96),(14,71),(15,94),(16,69),(17,92),(18,67),(19,90),(20,65),(21,88),(22,63),(23,86),(24,61),(25,84),(26,59),(27,82),(28,57),(29,80),(30,55),(31,78),(32,53),(33,76),(34,51),(35,74),(36,49),(37,72),(38,95),(39,70),(40,93),(41,68),(42,91),(43,66),(44,89),(45,64),(46,87),(47,62),(48,85)]])
81 conjugacy classes
| class | 1 | 2A | 2B | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 8A | 8B | 12A | ··· | 12H | 12I | 12J | 16A | 16B | 16C | 16D | 24A | ··· | 24P | 48A | ··· | 48AF |
| order | 1 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 12 | ··· | 12 | 12 | 12 | 16 | 16 | 16 | 16 | 24 | ··· | 24 | 48 | ··· | 48 |
| size | 1 | 1 | 24 | 1 | 1 | 2 | 2 | 2 | 2 | 24 | 1 | 1 | 2 | 2 | 2 | 24 | 24 | 2 | 2 | 2 | ··· | 2 | 24 | 24 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
81 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 |
| type | + | + | + | + | + | + | + | + | + | + | ||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | S3 | D4 | D6 | D8 | C3×S3 | D12 | C3×D4 | SD32 | S3×C6 | D24 | C3×D8 | C3×D12 | C48⋊C2 | C3×SD32 | C3×D24 | C3×C48⋊C2 |
| kernel | C3×C48⋊C2 | C3×C48 | C3×D24 | C3×Dic12 | C48⋊C2 | C48 | D24 | Dic12 | C48 | C3×C12 | C24 | C3×C6 | C16 | C12 | C12 | C32 | C8 | C6 | C6 | C4 | C3 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 4 | 2 | 4 | 4 | 4 | 8 | 8 | 8 | 16 |
Matrix representation of C3×C48⋊C2 ►in GL2(𝔽97) generated by
| 61 | 0 |
| 0 | 61 |
| 95 | 0 |
| 21 | 49 |
| 14 | 34 |
| 77 | 83 |
G:=sub<GL(2,GF(97))| [61,0,0,61],[95,21,0,49],[14,77,34,83] >;
C3×C48⋊C2 in GAP, Magma, Sage, TeX
C_3\times C_{48}\rtimes C_2 % in TeX
G:=Group("C3xC48:C2"); // GroupNames label
G:=SmallGroup(288,234);
// by ID
G=gap.SmallGroup(288,234);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,197,260,2355,80,2524,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^3=b^48=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^23>;
// generators/relations