direct product, metacyclic, supersoluble, monomial, A-group
Aliases: S3×C24, C24⋊4C6, D6.2C12, C12.64D6, Dic3.2C12, C3⋊C8⋊6C6, (C3×C24)⋊6C2, C3⋊1(C2×C24), C32⋊6(C2×C8), (S3×C6).4C4, (C4×S3).3C6, C2.1(S3×C12), C4.12(S3×C6), C6.21(C4×S3), C6.1(C2×C12), (S3×C12).6C2, C12.13(C2×C6), (C3×Dic3).4C4, (C3×C12).42C22, (C3×C3⋊C8)⋊13C2, (C3×C6).17(C2×C4), SmallGroup(144,69)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C24 |
Generators and relations for S3×C24
G = < a,b,c | a24=b3=c2=1, ab=ba, ac=ca, cbc=b-1 >
Smallest permutation representation of S3×C24
►On 48 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)
(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 41 33)(26 42 34)(27 43 35)(28 44 36)(29 45 37)(30 46 38)(31 47 39)(32 48 40)
(1 30)(2 31)(3 32)(4 33)(5 34)(6 35)(7 36)(8 37)(9 38)(10 39)(11 40)(12 41)(13 42)(14 43)(15 44)(16 45)(17 46)(18 47)(19 48)(20 25)(21 26)(22 27)(23 28)(24 29)
G:=sub<Sym(48)| (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), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,25)(21,26)(22,27)(23,28)(24,29)>;
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), (1,9,17)(2,10,18)(3,11,19)(4,12,20)(5,13,21)(6,14,22)(7,15,23)(8,16,24)(25,41,33)(26,42,34)(27,43,35)(28,44,36)(29,45,37)(30,46,38)(31,47,39)(32,48,40), (1,30)(2,31)(3,32)(4,33)(5,34)(6,35)(7,36)(8,37)(9,38)(10,39)(11,40)(12,41)(13,42)(14,43)(15,44)(16,45)(17,46)(18,47)(19,48)(20,25)(21,26)(22,27)(23,28)(24,29) );
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)], [(1,9,17),(2,10,18),(3,11,19),(4,12,20),(5,13,21),(6,14,22),(7,15,23),(8,16,24),(25,41,33),(26,42,34),(27,43,35),(28,44,36),(29,45,37),(30,46,38),(31,47,39),(32,48,40)], [(1,30),(2,31),(3,32),(4,33),(5,34),(6,35),(7,36),(8,37),(9,38),(10,39),(11,40),(12,41),(13,42),(14,43),(15,44),(16,45),(17,46),(18,47),(19,48),(20,25),(21,26),(22,27),(23,28),(24,29)]])
S3×C24 is a maximal subgroup of
C24.61D6 C24.63D6 C24.64D6 D6.1D12 D24⋊7S3 D6.3D12
72 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 | 24A | ··· | 24H | 24I | ··· | 24T | 24U | ··· | 24AB |
| 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 | 24 | ··· | 24 | 24 | ··· | 24 | 24 | ··· | 24 |
| 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 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | + | + | ||||||||||||||||
| image | C1 | C2 | C2 | C2 | C3 | C4 | C4 | C6 | C6 | C6 | C8 | C12 | C12 | C24 | S3 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C8 | S3×C12 | S3×C24 |
| kernel | S3×C24 | C3×C3⋊C8 | C3×C24 | S3×C12 | S3×C8 | C3×Dic3 | S3×C6 | C3⋊C8 | C24 | C4×S3 | C3×S3 | Dic3 | D6 | S3 | C24 | C12 | C8 | C6 | C4 | C3 | C2 | C1 |
| # reps | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 8 | 4 | 4 | 16 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 8 |
Matrix representation of S3×C24 ►in GL2(𝔽73) generated by
| 7 | 0 |
| 0 | 7 |
| 64 | 0 |
| 0 | 8 |
| 0 | 72 |
| 72 | 0 |
G:=sub<GL(2,GF(73))| [7,0,0,7],[64,0,0,8],[0,72,72,0] >;
S3×C24 in GAP, Magma, Sage, TeX
S_3\times C_{24} % in TeX
G:=Group("S3xC24"); // GroupNames label
G:=SmallGroup(144,69);
// by ID
G=gap.SmallGroup(144,69);
# by ID
G:=PCGroup([6,-2,-2,-3,-2,-2,-3,79,69,3461]);
// Polycyclic
G:=Group<a,b,c|a^24=b^3=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^-1>;
// generators/relations
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