direct product, metabelian, supersoluble, monomial
Aliases: C2×D6.3D6, C62.142C23, C3⋊D4⋊13D6, C23.29S32, C6⋊4(C4○D12), C6⋊3(D4⋊2S3), (C2×Dic3)⋊21D6, (C3×C6).29C24, C6.29(S3×C23), (C22×C6).99D6, (S3×C6).16C23, D6.16(C22×S3), (C22×S3).56D6, C32⋊7D4⋊8C22, C3⋊D12⋊10C22, (C22×Dic3)⋊11S3, (C6×Dic3)⋊29C22, (S3×Dic3)⋊16C22, C32⋊2Q8⋊15C22, C6.D6⋊11C22, C3⋊Dic3.27C23, (C2×C62).77C22, (C3×Dic3).23C23, Dic3.15(C22×S3), C3⋊5(C2×C4○D12), (C6×C3⋊D4)⋊4C2, C22.9(C2×S32), (C3×C6)⋊5(C4○D4), C3⋊4(C2×D4⋊2S3), (C2×C3⋊D4)⋊13S3, C2.30(C22×S32), (Dic3×C2×C6)⋊13C2, (C2×S3×Dic3)⋊24C2, C32⋊11(C2×C4○D4), (C2×C6.D6)⋊5C2, (S3×C2×C6).66C22, (C2×C3⋊D12)⋊15C2, (C2×C32⋊2Q8)⋊17C2, (C2×C3⋊S3).30C23, (C2×C32⋊7D4)⋊15C2, (C3×C3⋊D4)⋊10C22, (C2×C6).157(C22×S3), (C22×C3⋊S3).59C22, (C2×C3⋊Dic3).105C22, SmallGroup(288,970)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C2×D6.3D6
G = < a,b,c,d,e | a2=b6=c2=d6=1, e2=b3, ab=ba, ac=ca, ad=da, ae=ea, cbc=b-1, bd=db, be=eb, dcd-1=ece-1=b3c, ede-1=d-1 >
Subgroups: 1218 in 355 conjugacy classes, 116 normal (36 characteristic)
C1, C2, C2, C2, C3, C3, C4, C22, C22, C22, S3, C6, C6, C6, C2×C4, D4, Q8, C23, C23, C32, Dic3, Dic3, C12, D6, D6, C2×C6, C2×C6, C2×C6, C22×C4, C2×D4, C2×Q8, C4○D4, C3×S3, C3⋊S3, C3×C6, C3×C6, C3×C6, Dic6, C4×S3, D12, C2×Dic3, C2×Dic3, C2×Dic3, C3⋊D4, C3⋊D4, C2×C12, C3×D4, C22×S3, C22×S3, C22×C6, C22×C6, C2×C4○D4, C3×Dic3, C3⋊Dic3, S3×C6, S3×C6, C2×C3⋊S3, C2×C3⋊S3, C62, C62, C62, C2×Dic6, S3×C2×C4, C2×D12, C4○D12, D4⋊2S3, C22×Dic3, C22×Dic3, C2×C3⋊D4, C2×C3⋊D4, C22×C12, C6×D4, S3×Dic3, C6.D6, C3⋊D12, C32⋊2Q8, C6×Dic3, C6×Dic3, C3×C3⋊D4, C2×C3⋊Dic3, C32⋊7D4, S3×C2×C6, C22×C3⋊S3, C2×C62, C2×C4○D12, C2×D4⋊2S3, C2×S3×Dic3, D6.3D6, C2×C6.D6, C2×C3⋊D12, C2×C32⋊2Q8, Dic3×C2×C6, C6×C3⋊D4, C2×C32⋊7D4, C2×D6.3D6
Quotients: C1, C2, C22, S3, C23, D6, C4○D4, C24, C22×S3, C2×C4○D4, S32, C4○D12, D4⋊2S3, S3×C23, C2×S32, C2×C4○D12, C2×D4⋊2S3, D6.3D6, C22×S32, C2×D6.3D6
►On 48 points
(1 13)(2 14)(3 15)(4 16)(5 17)(6 18)(7 19)(8 20)(9 21)(10 22)(11 23)(12 24)(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 36)(2 35)(3 34)(4 33)(5 32)(6 31)(7 27)(8 26)(9 25)(10 30)(11 29)(12 28)(13 48)(14 47)(15 46)(16 45)(17 44)(18 43)(19 39)(20 38)(21 37)(22 42)(23 41)(24 40)
(1 9 5 7 3 11)(2 10 6 8 4 12)(13 21 17 19 15 23)(14 22 18 20 16 24)(25 35 27 31 29 33)(26 36 28 32 30 34)(37 47 39 43 41 45)(38 48 40 44 42 46)
(1 36 4 33)(2 31 5 34)(3 32 6 35)(7 30 10 27)(8 25 11 28)(9 26 12 29)(13 48 16 45)(14 43 17 46)(15 44 18 47)(19 42 22 39)(20 37 23 40)(21 38 24 41)
G:=sub<Sym(48)| (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40), (1,9,5,7,3,11)(2,10,6,8,4,12)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,35,27,31,29,33)(26,36,28,32,30,34)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,36,4,33)(2,31,5,34)(3,32,6,35)(7,30,10,27)(8,25,11,28)(9,26,12,29)(13,48,16,45)(14,43,17,46)(15,44,18,47)(19,42,22,39)(20,37,23,40)(21,38,24,41)>;
G:=Group( (1,13)(2,14)(3,15)(4,16)(5,17)(6,18)(7,19)(8,20)(9,21)(10,22)(11,23)(12,24)(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), (1,36)(2,35)(3,34)(4,33)(5,32)(6,31)(7,27)(8,26)(9,25)(10,30)(11,29)(12,28)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,39)(20,38)(21,37)(22,42)(23,41)(24,40), (1,9,5,7,3,11)(2,10,6,8,4,12)(13,21,17,19,15,23)(14,22,18,20,16,24)(25,35,27,31,29,33)(26,36,28,32,30,34)(37,47,39,43,41,45)(38,48,40,44,42,46), (1,36,4,33)(2,31,5,34)(3,32,6,35)(7,30,10,27)(8,25,11,28)(9,26,12,29)(13,48,16,45)(14,43,17,46)(15,44,18,47)(19,42,22,39)(20,37,23,40)(21,38,24,41) );
G=PermutationGroup([[(1,13),(2,14),(3,15),(4,16),(5,17),(6,18),(7,19),(8,20),(9,21),(10,22),(11,23),(12,24),(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)], [(1,36),(2,35),(3,34),(4,33),(5,32),(6,31),(7,27),(8,26),(9,25),(10,30),(11,29),(12,28),(13,48),(14,47),(15,46),(16,45),(17,44),(18,43),(19,39),(20,38),(21,37),(22,42),(23,41),(24,40)], [(1,9,5,7,3,11),(2,10,6,8,4,12),(13,21,17,19,15,23),(14,22,18,20,16,24),(25,35,27,31,29,33),(26,36,28,32,30,34),(37,47,39,43,41,45),(38,48,40,44,42,46)], [(1,36,4,33),(2,31,5,34),(3,32,6,35),(7,30,10,27),(8,25,11,28),(9,26,12,29),(13,48,16,45),(14,43,17,46),(15,44,18,47),(19,42,22,39),(20,37,23,40),(21,38,24,41)]])
54 conjugacy classes
| class | 1 | 2A | 2B | 2C | 2D | 2E | 2F | 2G | 2H | 2I | 3A | 3B | 3C | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 4I | 4J | 6A | ··· | 6J | 6K | ··· | 6S | 6T | 6U | 12A | ··· | 12H | 12I | 12J |
| order | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 3 | 3 | 3 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 4 | 6 | ··· | 6 | 6 | ··· | 6 | 6 | 6 | 12 | ··· | 12 | 12 | 12 |
| size | 1 | 1 | 1 | 1 | 2 | 2 | 6 | 6 | 18 | 18 | 2 | 2 | 4 | 3 | 3 | 3 | 3 | 6 | 6 | 6 | 6 | 18 | 18 | 2 | ··· | 2 | 4 | ··· | 4 | 12 | 12 | 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 | 4 | 4 | 4 | 4 |
| type | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | + | - | + | |||
| image | C1 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | C2 | S3 | S3 | D6 | D6 | D6 | D6 | C4○D4 | C4○D12 | S32 | D4⋊2S3 | C2×S32 | D6.3D6 |
| kernel | C2×D6.3D6 | C2×S3×Dic3 | D6.3D6 | C2×C6.D6 | C2×C3⋊D12 | C2×C32⋊2Q8 | Dic3×C2×C6 | C6×C3⋊D4 | C2×C32⋊7D4 | C22×Dic3 | C2×C3⋊D4 | C2×Dic3 | C3⋊D4 | C22×S3 | C22×C6 | C3×C6 | C6 | C23 | C6 | C22 | C2 |
| # reps | 1 | 1 | 8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 7 | 4 | 1 | 2 | 4 | 8 | 1 | 2 | 3 | 4 |
Matrix representation of C2×D6.3D6 ►in GL6(𝔽13)
| 1 | 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 | 12 | 0 |
| 0 | 0 | 0 | 0 | 0 | 12 |
| 12 | 1 | 0 | 0 | 0 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 12 | 0 | 0 | 0 |
| 0 | 0 | 0 | 12 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 1 | 0 | 0 | 0 | 0 | 0 |
| 1 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 5 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 12 |
| 0 | 0 | 0 | 0 | 1 | 0 |
| 12 | 0 | 0 | 0 | 0 | 0 |
| 0 | 12 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 8 | 0 | 0 |
| 0 | 0 | 8 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 12 | 0 |
| 0 | 0 | 0 | 0 | 12 | 1 |
G:=sub<GL(6,GF(13))| [1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,12,0,0,0,0,0,0,12],[12,12,0,0,0,0,1,0,0,0,0,0,0,0,12,0,0,0,0,0,0,12,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,1,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,5,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,12,0],[12,0,0,0,0,0,0,12,0,0,0,0,0,0,0,8,0,0,0,0,8,0,0,0,0,0,0,0,12,12,0,0,0,0,0,1] >;
C2×D6.3D6 in GAP, Magma, Sage, TeX
C_2\times D_6._3D_6
% in TeX
G:=Group("C2xD6.3D6"); // GroupNames label
G:=SmallGroup(288,970);
// by ID
G=gap.SmallGroup(288,970);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,-3,120,675,1356,9414]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^6=c^2=d^6=1,e^2=b^3,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,d*c*d^-1=e*c*e^-1=b^3*c,e*d*e^-1=d^-1>;
// generators/relations