direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C48, C48⋊5C6, D6.2C24, C24.97D6, Dic3.2C24, C3⋊C16⋊6C6, C3⋊1(C2×C48), (C3×C48)⋊7C2, C3⋊C8.3C12, (S3×C8).3C6, (S3×C6).4C8, C8.19(S3×C6), C2.1(S3×C24), C6.22(S3×C8), C6.1(C2×C24), C32⋊7(C2×C16), (S3×C24).6C2, (S3×C12).9C4, (C4×S3).4C12, C4.16(S3×C12), C24.32(C2×C6), C12.107(C4×S3), C12.21(C2×C12), (C3×Dic3).4C8, (C3×C24).64C22, (C3×C3⋊C8).6C4, (C3×C3⋊C16)⋊13C2, (C3×C6).27(C2×C8), (C3×C12).101(C2×C4), SmallGroup(288,231)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C48 |
Generators and relations for S3×C48
G = < a,b,c | a48=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Subgroups: 102 in 61 conjugacy classes, 38 normal (34 characteristic)
C1, C2, C2, C3, C3, C4, C4, C22, S3, C6, C6, C8, C8, C2×C4, C32, Dic3, C12, C12, D6, C2×C6, C16, C16, C2×C8, C3×S3, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×C12, C2×C16, C3×Dic3, C3×C12, S3×C6, C3⋊C16, C48, C48, S3×C8, C2×C24, C3×C3⋊C8, C3×C24, S3×C12, S3×C16, C2×C48, C3×C3⋊C16, C3×C48, S3×C24, S3×C48
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C12, D6, C2×C6, C16, C2×C8, C3×S3, C24, C4×S3, C2×C12, C2×C16, S3×C6, C48, S3×C8, C2×C24, S3×C12, S3×C16, C2×C48, S3×C24, S3×C48
►On 96 points
(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 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 52)(2 53)(3 54)(4 55)(5 56)(6 57)(7 58)(8 59)(9 60)(10 61)(11 62)(12 63)(13 64)(14 65)(15 66)(16 67)(17 68)(18 69)(19 70)(20 71)(21 72)(22 73)(23 74)(24 75)(25 76)(26 77)(27 78)(28 79)(29 80)(30 81)(31 82)(32 83)(33 84)(34 85)(35 86)(36 87)(37 88)(38 89)(39 90)(40 91)(41 92)(42 93)(43 94)(44 95)(45 96)(46 49)(47 50)(48 51)
G:=sub<Sym(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), (1,17,33)(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), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,49)(47,50)(48,51)>;
G:=Group( (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), (1,17,33)(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), (1,52)(2,53)(3,54)(4,55)(5,56)(6,57)(7,58)(8,59)(9,60)(10,61)(11,62)(12,63)(13,64)(14,65)(15,66)(16,67)(17,68)(18,69)(19,70)(20,71)(21,72)(22,73)(23,74)(24,75)(25,76)(26,77)(27,78)(28,79)(29,80)(30,81)(31,82)(32,83)(33,84)(34,85)(35,86)(36,87)(37,88)(38,89)(39,90)(40,91)(41,92)(42,93)(43,94)(44,95)(45,96)(46,49)(47,50)(48,51) );
G=PermutationGroup([[(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)], [(1,17,33),(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)], [(1,52),(2,53),(3,54),(4,55),(5,56),(6,57),(7,58),(8,59),(9,60),(10,61),(11,62),(12,63),(13,64),(14,65),(15,66),(16,67),(17,68),(18,69),(19,70),(20,71),(21,72),(22,73),(23,74),(24,75),(25,76),(26,77),(27,78),(28,79),(29,80),(30,81),(31,82),(32,83),(33,84),(34,85),(35,86),(36,87),(37,88),(38,89),(39,90),(40,91),(41,92),(42,93),(43,94),(44,95),(45,96),(46,49),(47,50),(48,51)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | 6B | 6C | 6D | 6E | 6F | 6G | 6H | 6I | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 16A | ··· | 16H | 16I | ··· | 16P | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB | 48A | ··· | 48P | 48Q | ··· | 48AN | 48AO | ··· | 48BD |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 16 | ··· | 16 | 16 | ··· | 16 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 | 48 | ··· | 48 | 48 | ··· | 48 | 48 | ··· | 48 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 1 | 1 | 2 | ··· | 2 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 |
144 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C8 | C12 | C12 | C16 | C24 | C24 | C48 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C12 | S3×C16 | S3×C24 | S3×C48 |
| kernel | S3×C48 | C3×C3⋊C16 | C3×C48 | S3×C24 | S3×C16 | C3×C3⋊C8 | S3×C12 | C3⋊C16 | C48 | S3×C8 | C3×Dic3 | S3×C6 | C3⋊C8 | C4×S3 | C3×S3 | Dic3 | D6 | S3 | C48 | C24 | C16 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 16 | 8 | 8 | 32 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 | 8 | 16 |
Matrix representation of S3×C48 ►in GL3(𝔽97) generated by
| 66 | 0 | 0 |
| 0 | 88 | 0 |
| 0 | 0 | 88 |
| 1 | 0 | 0 |
| 0 | 35 | 0 |
| 0 | 61 | 61 |
| 96 | 0 | 0 |
| 0 | 96 | 60 |
| 0 | 0 | 1 |
G:=sub<GL(3,GF(97))| [66,0,0,0,88,0,0,0,88],[1,0,0,0,35,61,0,0,61],[96,0,0,0,96,0,0,60,1] >;
S3×C48 in GAP, Magma, Sage, TeX
S_3\times C_{48} % in TeX
G:=Group("S3xC48"); // GroupNames label
G:=SmallGroup(288,231);
// by ID
G=gap.SmallGroup(288,231);
# by ID
G:=PCGroup([7,-2,-2,-3,-2,-2,-2,-3,92,80,102,9414]);
// Polycyclic
G:=Group<a,b,c|a^48=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations