direct product, metabelian, supersoluble, monomial
Aliases: C3×D6⋊D4, C62⋊9D4, C62.173C23, D6⋊C4⋊4C6, D6⋊4(C3×D4), C6.5(C6×D4), (S3×C6)⋊17D4, (C2×D12)⋊2C6, (C2×C12)⋊14D6, (C2×C6)⋊10D12, C2.7(C6×D12), (C6×D12)⋊25C2, (S3×C23)⋊4C6, C6.177(S3×D4), C6.93(C2×D12), C32⋊8C22≀C2, C22⋊4(C3×D12), (C6×C12)⋊18C22, C23.25(S3×C6), (C22×C6).106D6, (C6×Dic3)⋊17C22, (C2×C62).49C22, C2.7(C3×S3×D4), (C2×C4)⋊1(S3×C6), (C2×C6)⋊3(C3×D4), (C2×C12)⋊1(C2×C6), (C2×C3⋊D4)⋊1C6, (S3×C2×C6)⋊4C22, (S3×C22×C6)⋊3C2, (C3×D6⋊C4)⋊16C2, C3⋊1(C3×C22≀C2), (C6×C3⋊D4)⋊15C2, (C3×C22⋊C4)⋊3C6, C22⋊C4⋊2(C3×S3), C22.41(S3×C2×C6), (C3×C22⋊C4)⋊10S3, (C22×S3)⋊1(C2×C6), (C2×Dic3)⋊1(C2×C6), (C3×C6).206(C2×D4), (C2×C6).28(C22×C6), (C22×C6).23(C2×C6), (C32×C22⋊C4)⋊12C2, (C2×C6).306(C22×S3), SmallGroup(288,653)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C3×D6⋊D4
G = < a,b,c,d,e | a3=b6=c2=d4=e2=1, ab=ba, ac=ca, ad=da, ae=ea, cbc=ebe=b-1, bd=db, dcd-1=b3c, ece=bc, ede=d-1 >
Subgroups: 802 in 277 conjugacy classes, 74 normal (30 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, C2×C4, D4, C23, C23, C32, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22⋊C4, C22⋊C4, C2×D4, C24, C3×S3, C3×C6, C3×C6, C3×C6, D12, C2×Dic3, C3⋊D4, C2×C12, C2×C12, C3×D4, C22×S3, C22×S3, C22×S3, C22×C6, C22×C6, C22≀C2, C3×Dic3, C3×C12, S3×C6, S3×C6, C62, C62, C62, D6⋊C4, C3×C22⋊C4, C3×C22⋊C4, C2×D12, C2×C3⋊D4, C6×D4, S3×C23, C23×C6, C3×D12, C6×Dic3, C3×C3⋊D4, C6×C12, S3×C2×C6, S3×C2×C6, S3×C2×C6, C2×C62, D6⋊D4, C3×C22≀C2, C3×D6⋊C4, C32×C22⋊C4, C6×D12, C6×C3⋊D4, S3×C22×C6, C3×D6⋊D4
Quotients: C1, C2, C3, C22, S3, C6, D4, C23, D6, C2×C6, C2×D4, C3×S3, D12, C3×D4, C22×S3, C22×C6, C22≀C2, S3×C6, C2×D12, S3×D4, C6×D4, C3×D12, S3×C2×C6, D6⋊D4, C3×C22≀C2, C6×D12, C3×S3×D4, C3×D6⋊D4
►On 48 points
(1 3 5)(2 4 6)(7 11 9)(8 12 10)(13 15 17)(14 16 18)(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 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 33)(2 32)(3 31)(4 36)(5 35)(6 34)(7 29)(8 28)(9 27)(10 26)(11 25)(12 30)(13 39)(14 38)(15 37)(16 42)(17 41)(18 40)(19 46)(20 45)(21 44)(22 43)(23 48)(24 47)
(1 23 18 30)(2 24 13 25)(3 19 14 26)(4 20 15 27)(5 21 16 28)(6 22 17 29)(7 31 43 38)(8 32 44 39)(9 33 45 40)(10 34 46 41)(11 35 47 42)(12 36 48 37)
(1 10)(2 9)(3 8)(4 7)(5 12)(6 11)(13 45)(14 44)(15 43)(16 48)(17 47)(18 46)(19 39)(20 38)(21 37)(22 42)(23 41)(24 40)(25 33)(26 32)(27 31)(28 36)(29 35)(30 34)
G:=sub<Sym(48)| (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(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), (1,2,3,4,5,6)(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), (1,33)(2,32)(3,31)(4,36)(5,35)(6,34)(7,29)(8,28)(9,27)(10,26)(11,25)(12,30)(13,39)(14,38)(15,37)(16,42)(17,41)(18,40)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,23,18,30)(2,24,13,25)(3,19,14,26)(4,20,15,27)(5,21,16,28)(6,22,17,29)(7,31,43,38)(8,32,44,39)(9,33,45,40)(10,34,46,41)(11,35,47,42)(12,36,48,37), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34)>;
G:=Group( (1,3,5)(2,4,6)(7,11,9)(8,12,10)(13,15,17)(14,16,18)(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), (1,2,3,4,5,6)(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), (1,33)(2,32)(3,31)(4,36)(5,35)(6,34)(7,29)(8,28)(9,27)(10,26)(11,25)(12,30)(13,39)(14,38)(15,37)(16,42)(17,41)(18,40)(19,46)(20,45)(21,44)(22,43)(23,48)(24,47), (1,23,18,30)(2,24,13,25)(3,19,14,26)(4,20,15,27)(5,21,16,28)(6,22,17,29)(7,31,43,38)(8,32,44,39)(9,33,45,40)(10,34,46,41)(11,35,47,42)(12,36,48,37), (1,10)(2,9)(3,8)(4,7)(5,12)(6,11)(13,45)(14,44)(15,43)(16,48)(17,47)(18,46)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40)(25,33)(26,32)(27,31)(28,36)(29,35)(30,34) );
G=PermutationGroup([[(1,3,5),(2,4,6),(7,11,9),(8,12,10),(13,15,17),(14,16,18),(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)], [(1,2,3,4,5,6),(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)], [(1,33),(2,32),(3,31),(4,36),(5,35),(6,34),(7,29),(8,28),(9,27),(10,26),(11,25),(12,30),(13,39),(14,38),(15,37),(16,42),(17,41),(18,40),(19,46),(20,45),(21,44),(22,43),(23,48),(24,47)], [(1,23,18,30),(2,24,13,25),(3,19,14,26),(4,20,15,27),(5,21,16,28),(6,22,17,29),(7,31,43,38),(8,32,44,39),(9,33,45,40),(10,34,46,41),(11,35,47,42),(12,36,48,37)], [(1,10),(2,9),(3,8),(4,7),(5,12),(6,11),(13,45),(14,44),(15,43),(16,48),(17,47),(18,46),(19,39),(20,38),(21,37),(22,42),(23,41),(24,40),(25,33),(26,32),(27,31),(28,36),(29,35),(30,34)]])
72 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 2J | 3A | 3B | 3C | 3D | 3E | 4A | 4B | 4C | 6A | ··· | 6F | 6G | ··· | 6S | 6T | ··· | 6Y | 6Z | ··· | 6AG | 6AH | 6AI | 12A | ··· | 12P | 12Q | 12R |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 3 | 3 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 6 | 6 | 12 | 1 | 1 | 2 | 2 | 2 | 4 | 4 | 12 | 1 | ··· | 1 | 2 | ··· | 2 | 4 | ··· | 4 | 6 | ··· | 6 | 12 | 12 | 4 | ··· | 4 | 12 | 12 |
72 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | |||||||||||||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C3 | C6 | C6 | C6 | C6 | C6 | S3 | D4 | D4 | D6 | D6 | C3×S3 | C3×D4 | D12 | C3×D4 | S3×C6 | S3×C6 | C3×D12 | S3×D4 | C3×S3×D4 |
| kernel | C3×D6⋊D4 | C3×D6⋊C4 | C32×C22⋊C4 | C6×D12 | C6×C3⋊D4 | S3×C22×C6 | D6⋊D4 | D6⋊C4 | C3×C22⋊C4 | C2×D12 | C2×C3⋊D4 | S3×C23 | C3×C22⋊C4 | S3×C6 | C62 | C2×C12 | C22×C6 | C22⋊C4 | D6 | C2×C6 | C2×C6 | C2×C4 | C23 | C22 | C6 | C2 |
| # reps | 1 | 2 | 1 | 2 | 1 | 1 | 2 | 4 | 2 | 4 | 2 | 2 | 1 | 4 | 2 | 2 | 1 | 2 | 8 | 4 | 4 | 4 | 2 | 8 | 2 | 4 |
Matrix representation of C3×D6⋊D4 ►in GL6(𝔽13)
| 9 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 9 | 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 |
| 0 | 9 | 0 | 0 | 0 | 0 |
| 3 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 12 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 2 | 7 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 11 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 2 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 7 | 11 |
| 0 | 0 | 0 | 0 | 11 | 6 |
G:=sub<GL(6,GF(13))| [9,0,0,0,0,0,0,9,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,9,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[0,3,0,0,0,0,9,0,0,0,0,0,0,0,1,12,0,0,0,0,0,12,0,0,0,0,0,0,2,7,0,0,0,0,7,11],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,11,1,0,0,0,0,0,0,0,12,0,0,0,0,1,0],[0,12,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,0,2,12,0,0,0,0,0,0,7,11,0,0,0,0,11,6] >;
C3×D6⋊D4 in GAP, Magma, Sage, TeX
C_3\times D_6\rtimes D_4
% in TeX
G:=Group("C3xD6:D4"); // GroupNames label
G:=SmallGroup(288,653);
// by ID
G=gap.SmallGroup(288,653);
# by ID
G:=PCGroup([7,-2,-2,-2,-3,-2,-2,-3,590,555,142,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^3=b^6=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=e*b*e=b^-1,b*d=d*b,d*c*d^-1=b^3*c,e*c*e=b*c,e*d*e=d^-1>;
// generators/relations