direct product, metabelian, soluble, monomial, A-group
Aliases: C2×C42⋊C9, C42⋊3C18, (C2×C42)⋊C9, (C4×C12).3C6, C6.2(C42⋊C3), (C22×C6).8A4, C23.4(C3.A4), (C2×C4×C12).C3, C3.(C2×C42⋊C3), (C2×C6).5(C2×A4), C22.1(C2×C3.A4), SmallGroup(288,71)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C2×C42⋊C9 |
Generators and relations for C2×C42⋊C9
G = < a,b,c,d | a2=b4=c4=d9=1, ab=ba, ac=ca, ad=da, bc=cb, dbd-1=bc-1, dcd-1=b-1c2 >
Smallest permutation representation of C2×C42⋊C9
►On 36 points
Generators in S36
(1 12)(2 13)(3 14)(4 15)(5 16)(6 17)(7 18)(8 10)(9 11)(19 28)(20 29)(21 30)(22 31)(23 32)(24 33)(25 34)(26 35)(27 36)
(1 23 12 32)(2 13)(3 25 14 34)(4 26 15 35)(5 16)(6 19 17 28)(7 20 18 29)(8 10)(9 22 11 31)(21 30)(24 33)(27 36)
(1 32 12 23)(2 24 13 33)(4 35 15 26)(5 27 16 36)(7 29 18 20)(8 21 10 30)
(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)
G:=sub<Sym(36)| (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,23,12,32)(2,13)(3,25,14,34)(4,26,15,35)(5,16)(6,19,17,28)(7,20,18,29)(8,10)(9,22,11,31)(21,30)(24,33)(27,36), (1,32,12,23)(2,24,13,33)(4,35,15,26)(5,27,16,36)(7,29,18,20)(8,21,10,30), (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)>;
G:=Group( (1,12)(2,13)(3,14)(4,15)(5,16)(6,17)(7,18)(8,10)(9,11)(19,28)(20,29)(21,30)(22,31)(23,32)(24,33)(25,34)(26,35)(27,36), (1,23,12,32)(2,13)(3,25,14,34)(4,26,15,35)(5,16)(6,19,17,28)(7,20,18,29)(8,10)(9,22,11,31)(21,30)(24,33)(27,36), (1,32,12,23)(2,24,13,33)(4,35,15,26)(5,27,16,36)(7,29,18,20)(8,21,10,30), (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) );
G=PermutationGroup([[(1,12),(2,13),(3,14),(4,15),(5,16),(6,17),(7,18),(8,10),(9,11),(19,28),(20,29),(21,30),(22,31),(23,32),(24,33),(25,34),(26,35),(27,36)], [(1,23,12,32),(2,13),(3,25,14,34),(4,26,15,35),(5,16),(6,19,17,28),(7,20,18,29),(8,10),(9,22,11,31),(21,30),(24,33),(27,36)], [(1,32,12,23),(2,24,13,33),(4,35,15,26),(5,27,16,36),(7,29,18,20),(8,21,10,30)], [(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)]])
48 conjugacy classes
| class | 1 | 2A | 2B | 2C | 3A | 3B | 4A | ··· | 4H | 6A | 6B | 6C | 6D | 6E | 6F | 9A | ··· | 9F | 12A | ··· | 12P | 18A | ··· | 18F |
| order | 1 | 2 | 2 | 2 | 3 | 3 | 4 | ··· | 4 | 6 | 6 | 6 | 6 | 6 | 6 | 9 | ··· | 9 | 12 | ··· | 12 | 18 | ··· | 18 |
| size | 1 | 1 | 3 | 3 | 1 | 1 | 3 | ··· | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 16 | ··· | 16 | 3 | ··· | 3 | 16 | ··· | 16 |
48 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 |
| type | + | + | + | + | ||||||||||
| image | C1 | C2 | C3 | C6 | C9 | C18 | A4 | C2×A4 | C3.A4 | C42⋊C3 | C2×C3.A4 | C2×C42⋊C3 | C42⋊C9 | C2×C42⋊C9 |
| kernel | C2×C42⋊C9 | C42⋊C9 | C2×C4×C12 | C4×C12 | C2×C42 | C42 | C22×C6 | C2×C6 | C23 | C6 | C22 | C3 | C2 | C1 |
| # reps | 1 | 1 | 2 | 2 | 6 | 6 | 1 | 1 | 2 | 4 | 2 | 4 | 8 | 8 |
Matrix representation of C2×C42⋊C9 ►in GL3(𝔽13) generated by
| 12 | 0 | 0 |
| 0 | 12 | 0 |
| 0 | 0 | 12 |
| 8 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 1 |
| 12 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 5 |
| 0 | 0 | 9 |
| 12 | 0 | 0 |
| 0 | 12 | 0 |
G:=sub<GL(3,GF(13))| [12,0,0,0,12,0,0,0,12],[8,0,0,0,5,0,0,0,1],[12,0,0,0,5,0,0,0,5],[0,12,0,0,0,12,9,0,0] >;
C2×C42⋊C9 in GAP, Magma, Sage, TeX
C_2\times C_4^2\rtimes C_9
% in TeX
G:=Group("C2xC4^2:C9"); // GroupNames label
G:=SmallGroup(288,71);
// by ID
G=gap.SmallGroup(288,71);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,50,514,360,3476,102,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^2=b^4=c^4=d^9=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,d*b*d^-1=b*c^-1,d*c*d^-1=b^-1*c^2>;
// generators/relations
Export