direct product, metabelian, supersoluble, monomial
Aliases: S3×C6×D4, C62⋊4C23, C6⋊2(C6×D4), (C6×D4)⋊5C6, C12⋊(C22×C6), D12⋊7(C2×C6), (C2×C12)⋊24D6, C23⋊5(S3×C6), (C6×D12)⋊18C2, (C2×D12)⋊11C6, (S3×C23)⋊7C6, (S3×C6)⋊9C23, (C3×C12)⋊4C23, C12⋊6(C22×S3), (C6×C12)⋊9C22, D6⋊2(C22×C6), (C22×C6)⋊12D6, C6.5(C23×C6), C6.73(S3×C23), (C3×C6).42C24, (C2×C62)⋊9C22, C32⋊11(C22×D4), (S3×C12)⋊19C22, (C3×D12)⋊33C22, Dic3⋊1(C22×C6), (C3×Dic3)⋊9C23, (C6×Dic3)⋊33C22, (D4×C32)⋊18C22, C4⋊1(S3×C2×C6), C3⋊2(D4×C2×C6), (S3×C2×C4)⋊3C6, (D4×C3×C6)⋊9C2, (C2×C4)⋊6(S3×C6), C22⋊3(S3×C2×C6), (S3×C2×C12)⋊11C2, (C4×S3)⋊3(C2×C6), (C2×C12)⋊2(C2×C6), (C3×D4)⋊5(C2×C6), (C3×C6)⋊10(C2×D4), C3⋊D4⋊1(C2×C6), (C2×C3⋊D4)⋊9C6, (S3×C22×C6)⋊9C2, C2.6(S3×C22×C6), (C6×C3⋊D4)⋊23C2, (S3×C2×C6)⋊20C22, (C2×C6)⋊2(C22×C6), (C22×C6)⋊5(C2×C6), (C2×C6)⋊7(C22×S3), (C22×S3)⋊7(C2×C6), (C2×Dic3)⋊8(C2×C6), (C3×C3⋊D4)⋊15C22, SmallGroup(288,992)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×C6×D4
G = < a,b,c,d,e | a6=b3=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, cd=dc, ce=ec, ede=d-1 >
Subgroups: 1194 in 499 conjugacy classes, 194 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C4, C22, C22, C22, S3, S3, C6, C6, C6, C2×C4, C2×C4, D4, D4, C23, C23, C32, Dic3, C12, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×D4, C24, C3×S3, C3×S3, C3×C6, C3×C6, C3×C6, C4×S3, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C22×D4, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C22×C12, C6×D4, C6×D4, S3×C23, C23×C6, S3×C12, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, D4×C32, S3×C2×C6, S3×C2×C6, S3×C2×C6, C2×C62, C2×S3×D4, D4×C2×C6, S3×C2×C12, C6×D12, C3×S3×D4, C6×C3⋊D4, D4×C3×C6, S3×C22×C6, S3×C6×D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C24, C3×S3, C3×D4, C22×S3, C22×C6, C22×D4, S3×C6, S3×D4, C6×D4, S3×C23, C23×C6, S3×C2×C6, C2×S3×D4, D4×C2×C6, C3×S3×D4, S3×C22×C6, S3×C6×D4
►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 3 5)(2 4 6)(7 9 11)(8 10 12)(13 17 15)(14 18 16)(19 21 23)(20 22 24)(25 27 29)(26 28 30)(31 35 33)(32 36 34)(37 41 39)(38 42 40)(43 47 45)(44 48 46)
(1 35)(2 36)(3 31)(4 32)(5 33)(6 34)(7 38)(8 39)(9 40)(10 41)(11 42)(12 37)(13 30)(14 25)(15 26)(16 27)(17 28)(18 29)(19 46)(20 47)(21 48)(22 43)(23 44)(24 45)
(1 8 23 28)(2 9 24 29)(3 10 19 30)(4 11 20 25)(5 12 21 26)(6 7 22 27)(13 31 41 46)(14 32 42 47)(15 33 37 48)(16 34 38 43)(17 35 39 44)(18 36 40 45)
(7 27)(8 28)(9 29)(10 30)(11 25)(12 26)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)
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), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45), (1,8,23,28)(2,9,24,29)(3,10,19,30)(4,11,20,25)(5,12,21,26)(6,7,22,27)(13,31,41,46)(14,32,42,47)(15,33,37,48)(16,34,38,43)(17,35,39,44)(18,36,40,45), (7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)>;
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), (1,3,5)(2,4,6)(7,9,11)(8,10,12)(13,17,15)(14,18,16)(19,21,23)(20,22,24)(25,27,29)(26,28,30)(31,35,33)(32,36,34)(37,41,39)(38,42,40)(43,47,45)(44,48,46), (1,35)(2,36)(3,31)(4,32)(5,33)(6,34)(7,38)(8,39)(9,40)(10,41)(11,42)(12,37)(13,30)(14,25)(15,26)(16,27)(17,28)(18,29)(19,46)(20,47)(21,48)(22,43)(23,44)(24,45), (1,8,23,28)(2,9,24,29)(3,10,19,30)(4,11,20,25)(5,12,21,26)(6,7,22,27)(13,31,41,46)(14,32,42,47)(15,33,37,48)(16,34,38,43)(17,35,39,44)(18,36,40,45), (7,27)(8,28)(9,29)(10,30)(11,25)(12,26)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40) );
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)], [(1,3,5),(2,4,6),(7,9,11),(8,10,12),(13,17,15),(14,18,16),(19,21,23),(20,22,24),(25,27,29),(26,28,30),(31,35,33),(32,36,34),(37,41,39),(38,42,40),(43,47,45),(44,48,46)], [(1,35),(2,36),(3,31),(4,32),(5,33),(6,34),(7,38),(8,39),(9,40),(10,41),(11,42),(12,37),(13,30),(14,25),(15,26),(16,27),(17,28),(18,29),(19,46),(20,47),(21,48),(22,43),(23,44),(24,45)], [(1,8,23,28),(2,9,24,29),(3,10,19,30),(4,11,20,25),(5,12,21,26),(6,7,22,27),(13,31,41,46),(14,32,42,47),(15,33,37,48),(16,34,38,43),(17,35,39,44),(18,36,40,45)], [(7,27),(8,28),(9,29),(10,30),(11,25),(12,26),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 2L | 2M | 2N | 2O | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 4D | 6A | ··· | 6F | 6G | ··· | 6W | 6X | ··· | 6AE | 6AF | ··· | 6AQ | 6AR | ··· | 6AY | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | 1 | ··· | 1 | 2 | ··· | 2 | 3 | ··· | 3 | 4 | ··· | 4 | 6 | ··· | 6 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 |
90 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 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D6 | D6 | D6 | C3×S3 | C3×D4 | S3×C6 | S3×C6 | S3×C6 | S3×D4 | C3×S3×D4 |
| kernel | S3×C6×D4 | S3×C2×C12 | C6×D12 | C3×S3×D4 | C6×C3⋊D4 | D4×C3×C6 | S3×C22×C6 | C2×S3×D4 | S3×C2×C4 | C2×D12 | S3×D4 | C2×C3⋊D4 | C6×D4 | S3×C23 | C6×D4 | S3×C6 | C2×C12 | C3×D4 | C22×C6 | C2×D4 | D6 | C2×C4 | D4 | C23 | C6 | C2 |
| # reps | 1 | 1 | 1 | 8 | 2 | 1 | 2 | 2 | 2 | 2 | 16 | 4 | 2 | 4 | 1 | 4 | 1 | 4 | 2 | 2 | 8 | 2 | 8 | 4 | 2 | 4 |
Matrix representation of S3×C6×D4 ►in GL4(𝔽13) generated by
| 4 | 0 | 0 | 0 |
| 0 | 4 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 3 | 0 | 0 | 0 |
| 9 | 9 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 1 |
| 1 | 5 | 0 | 0 |
| 0 | 12 | 0 | 0 |
| 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 12 | 2 |
| 0 | 0 | 12 | 1 |
| 1 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 12 |
G:=sub<GL(4,GF(13))| [4,0,0,0,0,4,0,0,0,0,12,0,0,0,0,12],[3,9,0,0,0,9,0,0,0,0,1,0,0,0,0,1],[1,0,0,0,5,12,0,0,0,0,12,0,0,0,0,12],[1,0,0,0,0,1,0,0,0,0,12,12,0,0,2,1],[1,0,0,0,0,1,0,0,0,0,1,1,0,0,0,12] >;
S3×C6×D4 in GAP, Magma, Sage, TeX
S_3\times C_6\times D_4
% in TeX
G:=Group("S3xC6xD4"); // GroupNames label
G:=SmallGroup(288,992);
// by ID
G=gap.SmallGroup(288,992);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-3,-2,-3,409,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^6=b^3=c^2=d^4=e^2=1,a*b=b*a,a*c=c*a,a*d=d*a,a*e=e*a,c*b*c=b^-1,b*d=d*b,b*e=e*b,c*d=d*c,c*e=e*c,e*d*e=d^-1>;
// generators/relations