direct product, metabelian, supersoluble, monomial
Aliases: S3×D6⋊C4, D6.14D12, C62.90C23, D6⋊8(C4×S3), (C2×C12)⋊19D6, C2.5(S3×D12), D6⋊Dic3⋊5C2, (C2×Dic3)⋊9D6, (S3×C6).24D4, C6.145(S3×D4), C6.27(C2×D12), (C6×C12)⋊19C22, D6.18(C3⋊D4), (C22×S3).44D6, C6.11D12⋊18C2, C6.D12⋊14C2, (C6×Dic3)⋊11C22, (C2×C4)⋊5S32, (C2×S32)⋊2C4, (S3×C2×C4)⋊14S3, C2.24(C4×S32), C3⋊1(C2×D6⋊C4), C6.23(S3×C2×C4), (S3×C6)⋊7(C2×C4), (S3×C2×C12)⋊21C2, C3⋊4(S3×C22⋊C4), C2.4(S3×C3⋊D4), (C3×D6⋊C4)⋊24C2, (C22×S32).1C2, C22.44(C2×S32), (C2×S3×Dic3)⋊17C2, C6.40(C2×C3⋊D4), C32⋊2(C2×C22⋊C4), (C3×C6).112(C2×D4), (S3×C2×C6).82C22, (C3×S3)⋊1(C22⋊C4), (C3×C6).22(C22×C4), (C2×C3⋊Dic3)⋊1C22, (C2×C6).109(C22×S3), (C22×C3⋊S3).26C22, (C2×C3⋊S3)⋊4(C2×C4), SmallGroup(288,568)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for S3×D6⋊C4
G = < a,b,c,d,e | a3=b2=c6=d2=e4=1, bab=a-1, ac=ca, ad=da, ae=ea, bc=cb, bd=db, be=eb, dcd=c-1, ce=ec, ede-1=c3d >
Subgroups: 1242 in 281 conjugacy classes, 74 normal (44 characteristic)
C1, C2, C2, C3, C3, C4, C22, C22, S3, S3, C6, C6, C2×C4, C2×C4, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C22⋊C4, C22×C4, C24, C3×S3, C3×S3, C3⋊S3, C3×C6, C4×S3, C2×Dic3, C2×Dic3, C2×C12, C2×C12, C22×S3, C22×S3, C22×C6, C2×C22⋊C4, C3×Dic3, C3⋊Dic3, C3×C12, S32, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, D6⋊C4, D6⋊C4, C6.D4, C3×C22⋊C4, S3×C2×C4, S3×C2×C4, C22×Dic3, C22×C12, S3×C23, S3×Dic3, S3×C12, C6×Dic3, C2×C3⋊Dic3, C6×C12, C2×S32, C2×S32, S3×C2×C6, C22×C3⋊S3, S3×C22⋊C4, C2×D6⋊C4, D6⋊Dic3, C6.D12, C3×D6⋊C4, C6.11D12, C2×S3×Dic3, S3×C2×C12, C22×S32, S3×D6⋊C4
Quotients: C1, C2, C4, C22, S3, C2×C4, D4, C23, D6, C22⋊C4, C22×C4, C2×D4, C4×S3, D12, C3⋊D4, C22×S3, C2×C22⋊C4, S32, D6⋊C4, S3×C2×C4, C2×D12, S3×D4, C2×C3⋊D4, C2×S32, S3×C22⋊C4, C2×D6⋊C4, C4×S32, S3×D12, S3×C3⋊D4, S3×D6⋊C4
►On 48 points
(1 5 3)(2 6 4)(7 11 9)(8 12 10)(13 17 15)(14 18 16)(19 23 21)(20 24 22)(25 27 29)(26 28 30)(31 33 35)(32 34 36)(37 39 41)(38 40 42)(43 45 47)(44 46 48)
(1 26)(2 27)(3 28)(4 29)(5 30)(6 25)(7 32)(8 33)(9 34)(10 35)(11 36)(12 31)(13 41)(14 42)(15 37)(16 38)(17 39)(18 40)(19 47)(20 48)(21 43)(22 44)(23 45)(24 46)
(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 28)(2 27)(3 26)(4 25)(5 30)(6 29)(7 34)(8 33)(9 32)(10 31)(11 36)(12 35)(13 40)(14 39)(15 38)(16 37)(17 42)(18 41)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 19 7 13)(2 20 8 14)(3 21 9 15)(4 22 10 16)(5 23 11 17)(6 24 12 18)(25 46 31 40)(26 47 32 41)(27 48 33 42)(28 43 34 37)(29 44 35 38)(30 45 36 39)
G:=sub<Sym(48)| (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,47)(20,48)(21,43)(22,44)(23,45)(24,46), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,34)(8,33)(9,32)(10,31)(11,36)(12,35)(13,40)(14,39)(15,38)(16,37)(17,42)(18,41)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39)>;
G:=Group( (1,5,3)(2,6,4)(7,11,9)(8,12,10)(13,17,15)(14,18,16)(19,23,21)(20,24,22)(25,27,29)(26,28,30)(31,33,35)(32,34,36)(37,39,41)(38,40,42)(43,45,47)(44,46,48), (1,26)(2,27)(3,28)(4,29)(5,30)(6,25)(7,32)(8,33)(9,34)(10,35)(11,36)(12,31)(13,41)(14,42)(15,37)(16,38)(17,39)(18,40)(19,47)(20,48)(21,43)(22,44)(23,45)(24,46), (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), (1,28)(2,27)(3,26)(4,25)(5,30)(6,29)(7,34)(8,33)(9,32)(10,31)(11,36)(12,35)(13,40)(14,39)(15,38)(16,37)(17,42)(18,41)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,19,7,13)(2,20,8,14)(3,21,9,15)(4,22,10,16)(5,23,11,17)(6,24,12,18)(25,46,31,40)(26,47,32,41)(27,48,33,42)(28,43,34,37)(29,44,35,38)(30,45,36,39) );
G=PermutationGroup([[(1,5,3),(2,6,4),(7,11,9),(8,12,10),(13,17,15),(14,18,16),(19,23,21),(20,24,22),(25,27,29),(26,28,30),(31,33,35),(32,34,36),(37,39,41),(38,40,42),(43,45,47),(44,46,48)], [(1,26),(2,27),(3,28),(4,29),(5,30),(6,25),(7,32),(8,33),(9,34),(10,35),(11,36),(12,31),(13,41),(14,42),(15,37),(16,38),(17,39),(18,40),(19,47),(20,48),(21,43),(22,44),(23,45),(24,46)], [(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)], [(1,28),(2,27),(3,26),(4,25),(5,30),(6,29),(7,34),(8,33),(9,32),(10,31),(11,36),(12,35),(13,40),(14,39),(15,38),(16,37),(17,42),(18,41),(19,46),(20,45),(21,44),(22,43),(23,48),(24,47)], [(1,19,7,13),(2,20,8,14),(3,21,9,15),(4,22,10,16),(5,23,11,17),(6,24,12,18),(25,46,31,40),(26,47,32,41),(27,48,33,42),(28,43,34,37),(29,44,35,38),(30,45,36,39)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 2K | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6A | ··· | 6F | 6G | 6H | 6I | 6J | 6K | 6L | 6M | 6N | 6O | 12A | 12B | 12C | 12D | 12E | ··· | 12J | 12K | 12L | 12M | 12N | 12O | 12P |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 6 | 12 | 12 | 12 | 12 | 12 | ··· | 12 | 12 | 12 | 12 | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 2 | 2 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | ··· | 2 | 4 | 4 | 4 | 6 | 6 | 6 | 6 | 12 | 12 | 2 | 2 | 2 | 2 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 12 | 12 |
54 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C4 | S3 | S3 | D4 | D6 | D6 | D6 | C4×S3 | D12 | C3⋊D4 | S32 | S3×D4 | C2×S32 | C4×S32 | S3×D12 | S3×C3⋊D4 |
| kernel | S3×D6⋊C4 | D6⋊Dic3 | C6.D12 | C3×D6⋊C4 | C6.11D12 | C2×S3×Dic3 | S3×C2×C12 | C22×S32 | C2×S32 | D6⋊C4 | S3×C2×C4 | S3×C6 | C2×Dic3 | C2×C12 | C22×S3 | D6 | D6 | D6 | C2×C4 | C6 | C22 | C2 | C2 | C2 |
| # reps | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 8 | 1 | 1 | 4 | 2 | 2 | 2 | 8 | 4 | 4 | 1 | 2 | 1 | 2 | 2 | 2 |
Matrix representation of S3×D6⋊C4 ►in GL6(𝔽13)
| 0 | 1 | 0 | 0 | 0 | 0 |
| 12 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 1 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 5 | 10 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 0 | 0 | 5 | 0 |
| 0 | 0 | 0 | 0 | 0 | 5 |
G:=sub<GL(6,GF(13))| [0,12,0,0,0,0,1,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,12,12],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,1,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,1,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,5,0,0,0,0,0,10,8,0,0,0,0,0,0,5,0,0,0,0,0,0,5] >;
S3×D6⋊C4 in GAP, Magma, Sage, TeX
S_3\times D_6\rtimes C_4
% in TeX
G:=Group("S3xD6:C4"); // GroupNames label
G:=SmallGroup(288,568);
// by ID
G=gap.SmallGroup(288,568);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,219,58,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^2=c^6=d^2=e^4=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,a*e=e*a,b*c=c*b,b*d=d*b,b*e=e*b,d*c*d=c^-1,c*e=e*c,e*d*e^-1=c^3*d>;
// generators/relations