direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C24, C6⋊1(C2×C24), (C6×C24)⋊15C2, (C2×C24)⋊10C6, C24⋊11(C2×C6), (C4×S3).5C12, C4.23(S3×C12), C3⋊1(C22×C24), D6.8(C2×C12), C32⋊7(C22×C8), (S3×C12).10C4, C12.114(C4×S3), C12.28(C2×C12), (C3×C24)⋊29C22, (C2×C12).460D6, C62.76(C2×C4), (C22×S3).5C12, C6.12(C22×C12), C12.35(C22×C6), C22.13(S3×C12), (C2×Dic3).8C12, (C6×Dic3).16C4, (S3×C12).61C22, (C6×C12).347C22, C12.223(C22×S3), (C3×C12).167C23, Dic3.10(C2×C12), (C6×C3⋊C8)⋊27C2, (C2×C3⋊C8)⋊13C6, C3⋊C8⋊13(C2×C6), (C3×C6)⋊6(C2×C8), C2.2(S3×C2×C12), C4.35(S3×C2×C6), (S3×C2×C4).12C6, (S3×C2×C6).10C4, C6.111(S3×C2×C4), (C3×C3⋊C8)⋊44C22, (S3×C2×C12).25C2, (C2×C6).83(C4×S3), (C2×C4).98(S3×C6), (S3×C6).27(C2×C4), (C4×S3).18(C2×C6), (C2×C6).17(C2×C12), (C2×C12).128(C2×C6), (C3×C12).113(C2×C4), (C3×C6).83(C22×C4), (C3×Dic3).31(C2×C4), SmallGroup(288,670)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C24 |
Generators and relations for S3×C2×C24
G = < a,b,c,d | a2=b24=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 282 in 163 conjugacy classes, 98 normal (38 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C8, C8, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C2×C8, C2×C8, C22×C4, C3×S3, C3×C6, C3×C6, C3⋊C8, C24, C24, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C22×C8, C3×Dic3, C3×C12, S3×C6, C62, S3×C8, C2×C3⋊C8, C2×C24, C2×C24, S3×C2×C4, C22×C12, C3×C3⋊C8, C3×C24, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C8, C22×C24, S3×C24, C6×C3⋊C8, C6×C24, S3×C2×C12, S3×C2×C24
Quotients: C1, C2, C3, C4, C22, S3, C6, C8, C2×C4, C23, C12, D6, C2×C6, C2×C8, C22×C4, C3×S3, C24, C4×S3, C2×C12, C22×S3, C22×C6, C22×C8, S3×C6, S3×C8, C2×C24, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, S3×C2×C8, C22×C24, S3×C24, S3×C2×C12, S3×C2×C24
►On 96 points
(1 39)(2 40)(3 41)(4 42)(5 43)(6 44)(7 45)(8 46)(9 47)(10 48)(11 25)(12 26)(13 27)(14 28)(15 29)(16 30)(17 31)(18 32)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(49 92)(50 93)(51 94)(52 95)(53 96)(54 73)(55 74)(56 75)(57 76)(58 77)(59 78)(60 79)(61 80)(62 81)(63 82)(64 83)(65 84)(66 85)(67 86)(68 87)(69 88)(70 89)(71 90)(72 91)
(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 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 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 64)(26 65)(27 66)(28 67)(29 68)(30 69)(31 70)(32 71)(33 72)(34 49)(35 50)(36 51)(37 52)(38 53)(39 54)(40 55)(41 56)(42 57)(43 58)(44 59)(45 60)(46 61)(47 62)(48 63)
G:=sub<Sym(96)| (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(49,92)(50,93)(51,94)(52,95)(53,96)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (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), (1,9,17)(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), (1,73)(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)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63)>;
G:=Group( (1,39)(2,40)(3,41)(4,42)(5,43)(6,44)(7,45)(8,46)(9,47)(10,48)(11,25)(12,26)(13,27)(14,28)(15,29)(16,30)(17,31)(18,32)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(49,92)(50,93)(51,94)(52,95)(53,96)(54,73)(55,74)(56,75)(57,76)(58,77)(59,78)(60,79)(61,80)(62,81)(63,82)(64,83)(65,84)(66,85)(67,86)(68,87)(69,88)(70,89)(71,90)(72,91), (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), (1,9,17)(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), (1,73)(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)(25,64)(26,65)(27,66)(28,67)(29,68)(30,69)(31,70)(32,71)(33,72)(34,49)(35,50)(36,51)(37,52)(38,53)(39,54)(40,55)(41,56)(42,57)(43,58)(44,59)(45,60)(46,61)(47,62)(48,63) );
G=PermutationGroup([[(1,39),(2,40),(3,41),(4,42),(5,43),(6,44),(7,45),(8,46),(9,47),(10,48),(11,25),(12,26),(13,27),(14,28),(15,29),(16,30),(17,31),(18,32),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(49,92),(50,93),(51,94),(52,95),(53,96),(54,73),(55,74),(56,75),(57,76),(58,77),(59,78),(60,79),(61,80),(62,81),(63,82),(64,83),(65,84),(66,85),(67,86),(68,87),(69,88),(70,89),(71,90),(72,91)], [(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)], [(1,9,17),(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)], [(1,73),(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),(25,64),(26,65),(27,66),(28,67),(29,68),(30,69),(31,70),(32,71),(33,72),(34,49),(35,50),(36,51),(37,52),(38,53),(39,54),(40,55),(41,56),(42,57),(43,58),(44,59),(45,60),(46,61),(47,62),(48,63)]])
144 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | ··· | 6O | 6P | ··· | 6W | 8A | ··· | 8H | 8I | ··· | 8P | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB | 24A | ··· | 24P | 24Q | ··· | 24AN | 24AO | ··· | 24BD |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 8 | ··· | 8 | 8 | ··· | 8 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | 1 | 2 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 1 | ··· | 1 | 2 | ··· | 2 | 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 | 2 | 2 |
| type | + | + | + | + | + | + | + | + | ||||||||||||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C3 | C4 | C4 | C4 | C6 | C6 | C6 | C6 | C8 | C12 | C12 | C12 | C24 | S3 | D6 | D6 | C3×S3 | C4×S3 | C4×S3 | S3×C6 | S3×C6 | S3×C8 | S3×C12 | S3×C12 | S3×C24 |
| kernel | S3×C2×C24 | S3×C24 | C6×C3⋊C8 | C6×C24 | S3×C2×C12 | S3×C2×C8 | S3×C12 | C6×Dic3 | S3×C2×C6 | S3×C8 | C2×C3⋊C8 | C2×C24 | S3×C2×C4 | S3×C6 | C4×S3 | C2×Dic3 | C22×S3 | D6 | C2×C24 | C24 | C2×C12 | C2×C8 | C12 | C2×C6 | C8 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 4 | 2 | 2 | 8 | 2 | 2 | 2 | 16 | 8 | 4 | 4 | 32 | 1 | 2 | 1 | 2 | 2 | 2 | 4 | 2 | 8 | 4 | 4 | 16 |
Matrix representation of S3×C2×C24 ►in GL4(𝔽73) generated by
| 1 | 0 | 0 | 0 |
| 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 21 | 0 | 0 | 0 |
| 0 | 49 | 0 | 0 |
| 0 | 0 | 70 | 0 |
| 0 | 0 | 0 | 70 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 64 | 0 |
| 0 | 0 | 8 | 8 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 72 | 7 |
| 0 | 0 | 0 | 1 |
G:=sub<GL(4,GF(73))| [1,0,0,0,0,72,0,0,0,0,1,0,0,0,0,1],[21,0,0,0,0,49,0,0,0,0,70,0,0,0,0,70],[1,0,0,0,0,1,0,0,0,0,64,8,0,0,0,8],[1,0,0,0,0,1,0,0,0,0,72,0,0,0,7,1] >;
S3×C2×C24 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{24} % in TeX
G:=Group("S3xC2xC24"); // GroupNames label
G:=SmallGroup(288,670);
// by ID
G=gap.SmallGroup(288,670);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,142,102,9414]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^24=c^3=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations