direct product, metabelian, supersoluble, monomial, A-group
Aliases: S3×C2×C12, C62.27C22, C6⋊1(C2×C12), C12⋊3(C2×C6), (C2×C12)⋊5C6, (C6×C12)⋊8C2, D6.4(C2×C6), (C2×C6).47D6, C3⋊1(C22×C12), (C3×C12)⋊8C22, Dic3⋊3(C2×C6), (C2×Dic3)⋊5C6, C22.9(S3×C6), C6.2(C22×C6), C32⋊5(C22×C4), (C6×Dic3)⋊11C2, (C22×S3).2C6, (C3×C6).20C23, C6.41(C22×S3), (S3×C6).13C22, (C3×Dic3)⋊10C22, C2.1(S3×C2×C6), (C3×C6)⋊4(C2×C4), (S3×C2×C6).4C2, (C2×C6).12(C2×C6), SmallGroup(144,159)
Series: Derived ►Chief ►Lower central ►Upper central
| C3 — S3×C2×C12 |
Generators and relations for S3×C2×C12
G = < a,b,c,d | a2=b12=c3=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >
Subgroups: 200 in 116 conjugacy classes, 70 normal (22 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, C12, D6, C2×C6, C2×C6, C22×C4, C3×S3, C3×C6, C3×C6, C4×S3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×C6, C3×Dic3, C3×C12, S3×C6, C62, S3×C2×C4, C22×C12, S3×C12, C6×Dic3, C6×C12, S3×C2×C6, S3×C2×C12
Quotients: C1, C2, C3, C4, C22, S3, C6, C2×C4, C23, C12, D6, C2×C6, C22×C4, C3×S3, C4×S3, C2×C12, C22×S3, C22×C6, S3×C6, S3×C2×C4, C22×C12, S3×C12, S3×C2×C6, S3×C2×C12
►On 48 points
(1 19)(2 20)(3 21)(4 22)(5 23)(6 24)(7 13)(8 14)(9 15)(10 16)(11 17)(12 18)(25 37)(26 38)(27 39)(28 40)(29 41)(30 42)(31 43)(32 44)(33 45)(34 46)(35 47)(36 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 5 9)(2 6 10)(3 7 11)(4 8 12)(13 17 21)(14 18 22)(15 19 23)(16 20 24)(25 33 29)(26 34 30)(27 35 31)(28 36 32)(37 45 41)(38 46 42)(39 47 43)(40 48 44)
(1 43)(2 44)(3 45)(4 46)(5 47)(6 48)(7 37)(8 38)(9 39)(10 40)(11 41)(12 42)(13 25)(14 26)(15 27)(16 28)(17 29)(18 30)(19 31)(20 32)(21 33)(22 34)(23 35)(24 36)
G:=sub<Sym(48)| (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36)>;
G:=Group( (1,19)(2,20)(3,21)(4,22)(5,23)(6,24)(7,13)(8,14)(9,15)(10,16)(11,17)(12,18)(25,37)(26,38)(27,39)(28,40)(29,41)(30,42)(31,43)(32,44)(33,45)(34,46)(35,47)(36,48), (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), (1,5,9)(2,6,10)(3,7,11)(4,8,12)(13,17,21)(14,18,22)(15,19,23)(16,20,24)(25,33,29)(26,34,30)(27,35,31)(28,36,32)(37,45,41)(38,46,42)(39,47,43)(40,48,44), (1,43)(2,44)(3,45)(4,46)(5,47)(6,48)(7,37)(8,38)(9,39)(10,40)(11,41)(12,42)(13,25)(14,26)(15,27)(16,28)(17,29)(18,30)(19,31)(20,32)(21,33)(22,34)(23,35)(24,36) );
G=PermutationGroup([[(1,19),(2,20),(3,21),(4,22),(5,23),(6,24),(7,13),(8,14),(9,15),(10,16),(11,17),(12,18),(25,37),(26,38),(27,39),(28,40),(29,41),(30,42),(31,43),(32,44),(33,45),(34,46),(35,47),(36,48)], [(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)], [(1,5,9),(2,6,10),(3,7,11),(4,8,12),(13,17,21),(14,18,22),(15,19,23),(16,20,24),(25,33,29),(26,34,30),(27,35,31),(28,36,32),(37,45,41),(38,46,42),(39,47,43),(40,48,44)], [(1,43),(2,44),(3,45),(4,46),(5,47),(6,48),(7,37),(8,38),(9,39),(10,40),(11,41),(12,42),(13,25),(14,26),(15,27),(16,28),(17,29),(18,30),(19,31),(20,32),(21,33),(22,34),(23,35),(24,36)]])
S3×C2×C12 is a maximal subgroup of
C12.77D12 C62.11C23 C62.20C23 D6⋊Dic6 C62.25C23 D6⋊6Dic6 D6⋊7Dic6 C62.49C23 C62.74C23 C62.75C23 D6⋊D12 D6⋊2D12 C12⋊7D12
72 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 | 12A | ··· | 12H | 12I | ··· | 12T | 12U | ··· | 12AB |
| 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 | 12 | ··· | 12 | 12 | ··· | 12 | 12 | ··· | 12 |
| 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 | 2 | ··· | 2 | 3 | ··· | 3 |
72 irreducible representations
| dim | 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 | C2 | C3 | C4 | C6 | C6 | C6 | C6 | C12 | S3 | D6 | D6 | C3×S3 | C4×S3 | S3×C6 | S3×C6 | S3×C12 |
| kernel | S3×C2×C12 | S3×C12 | C6×Dic3 | C6×C12 | S3×C2×C6 | S3×C2×C4 | S3×C6 | C4×S3 | C2×Dic3 | C2×C12 | C22×S3 | D6 | C2×C12 | C12 | C2×C6 | C2×C4 | C6 | C4 | C22 | C2 |
| # reps | 1 | 4 | 1 | 1 | 1 | 2 | 8 | 8 | 2 | 2 | 2 | 16 | 1 | 2 | 1 | 2 | 4 | 4 | 2 | 8 |
Matrix representation of S3×C2×C12 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 4 | 0 | 0 |
| 0 | 2 | 0 |
| 0 | 0 | 2 |
| 1 | 0 | 0 |
| 0 | 3 | 3 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 11 | 1 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[4,0,0,0,2,0,0,0,2],[1,0,0,0,3,0,0,3,9],[12,0,0,0,12,11,0,0,1] >;
S3×C2×C12 in GAP, Magma, Sage, TeX
S_3\times C_2\times C_{12} % in TeX
G:=Group("S3xC2xC12"); // GroupNames label
G:=SmallGroup(144,159);
// by ID
G=gap.SmallGroup(144,159);
# by ID
G:=PCGroup([6,-2,-2,-2,-3,-2,-3,122,3461]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^12=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