non-abelian, soluble, monomial
Aliases: C42⋊D9, (C4×C12).S3, C42⋊C9⋊2C2, (C2×C6).2S4, C3.(C42⋊S3), C22.(C3.S4), SmallGroup(288,67)
Series: Derived ►Chief ►Lower central ►Upper central
| C42⋊C9 — C42⋊D9 |
Generators and relations for C42⋊D9
G = < a,b,c,d | a4=b4=c9=d2=1, ab=ba, cac-1=dbd=a-1b-1, ad=da, cbc-1=a, dcd=c-1 >
3C2
36C2
3C4
3C4
18C22
18C4
3C6
12S3
16C9
9D4
9Q8
18C2×C4
18C8
18D4
3C12
3C12
6D6
6Dic3
16D9
3Dic6
3D12
Character table of C42⋊D9
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 4D | 6 | 8A | 8B | 9A | 9B | 9C | 12A | 12B | 12C | 12D | |
| size | 1 | 3 | 36 | 2 | 3 | 3 | 6 | 36 | 6 | 36 | 36 | 32 | 32 | 32 | 6 | 6 | 6 | 6 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial |
| ρ2 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 2 |
| ρ3 | 2 | 2 | 0 | 2 | 2 | 2 | 2 | 0 | 2 | 0 | 0 | -1 | -1 | -1 | 2 | 2 | 2 | 2 | orthogonal lifted from S3 |
| ρ4 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | ζ98+ζ9 | ζ97+ζ92 | ζ95+ζ94 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ5 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | ζ95+ζ94 | ζ98+ζ9 | ζ97+ζ92 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ6 | 2 | 2 | 0 | -1 | 2 | 2 | 2 | 0 | -1 | 0 | 0 | ζ97+ζ92 | ζ95+ζ94 | ζ98+ζ9 | -1 | -1 | -1 | -1 | orthogonal lifted from D9 |
| ρ7 | 3 | 3 | -1 | 3 | -1 | -1 | -1 | -1 | 3 | 1 | 1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ8 | 3 | 3 | 1 | 3 | -1 | -1 | -1 | 1 | 3 | -1 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | orthogonal lifted from S4 |
| ρ9 | 3 | -1 | -1 | 3 | -1-2i | -1+2i | 1 | 1 | -1 | i | -i | 0 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | complex lifted from C42⋊S3 |
| ρ10 | 3 | -1 | 1 | 3 | -1+2i | -1-2i | 1 | -1 | -1 | i | -i | 0 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | complex lifted from C42⋊S3 |
| ρ11 | 3 | -1 | 1 | 3 | -1-2i | -1+2i | 1 | -1 | -1 | -i | i | 0 | 0 | 0 | -1-2i | 1 | 1 | -1+2i | complex lifted from C42⋊S3 |
| ρ12 | 3 | -1 | -1 | 3 | -1+2i | -1-2i | 1 | 1 | -1 | -i | i | 0 | 0 | 0 | -1+2i | 1 | 1 | -1-2i | complex lifted from C42⋊S3 |
| ρ13 | 6 | -2 | 0 | 6 | 2 | 2 | -2 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 2 | -2 | -2 | 2 | orthogonal lifted from C42⋊S3 |
| ρ14 | 6 | 6 | 0 | -3 | -2 | -2 | -2 | 0 | -3 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | orthogonal lifted from C3.S4 |
| ρ15 | 6 | -2 | 0 | -3 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | 1+2√3 | 1-2√3 | -1 | orthogonal faithful |
| ρ16 | 6 | -2 | 0 | -3 | 2 | 2 | -2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | -1 | 1-2√3 | 1+2√3 | -1 | orthogonal faithful |
| ρ17 | 6 | -2 | 0 | -3 | -2-4i | -2+4i | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1+2i | -1 | -1 | 1-2i | complex faithful |
| ρ18 | 6 | -2 | 0 | -3 | -2+4i | -2-4i | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 1-2i | -1 | -1 | 1+2i | complex faithful |
►On 36 points
Generators in S36
(1 34 19 10)(3 12 21 36)(4 28 22 13)(6 15 24 30)(7 31 25 16)(9 18 27 33)
(1 10 19 34)(2 35 20 11)(4 13 22 28)(5 29 23 14)(7 16 25 31)(8 32 26 17)
(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)
(1 33)(2 32)(3 31)(4 30)(5 29)(6 28)(7 36)(8 35)(9 34)(10 27)(11 26)(12 25)(13 24)(14 23)(15 22)(16 21)(17 20)(18 19)
G:=sub<Sym(36)| (1,34,19,10)(3,12,21,36)(4,28,22,13)(6,15,24,30)(7,31,25,16)(9,18,27,33), (1,10,19,34)(2,35,20,11)(4,13,22,28)(5,29,23,14)(7,16,25,31)(8,32,26,17), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,36)(8,35)(9,34)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)>;
G:=Group( (1,34,19,10)(3,12,21,36)(4,28,22,13)(6,15,24,30)(7,31,25,16)(9,18,27,33), (1,10,19,34)(2,35,20,11)(4,13,22,28)(5,29,23,14)(7,16,25,31)(8,32,26,17), (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), (1,33)(2,32)(3,31)(4,30)(5,29)(6,28)(7,36)(8,35)(9,34)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19) );
G=PermutationGroup([[(1,34,19,10),(3,12,21,36),(4,28,22,13),(6,15,24,30),(7,31,25,16),(9,18,27,33)], [(1,10,19,34),(2,35,20,11),(4,13,22,28),(5,29,23,14),(7,16,25,31),(8,32,26,17)], [(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)], [(1,33),(2,32),(3,31),(4,30),(5,29),(6,28),(7,36),(8,35),(9,34),(10,27),(11,26),(12,25),(13,24),(14,23),(15,22),(16,21),(17,20),(18,19)]])
Matrix representation of C42⋊D9 ►in GL5(𝔽73)
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 27 |
| 1 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 27 |
| 28 | 70 | 0 | 0 | 0 |
| 3 | 31 | 0 | 0 | 0 |
| 0 | 0 | 0 | 13 | 0 |
| 0 | 0 | 0 | 0 | 72 |
| 0 | 0 | 28 | 0 | 0 |
| 28 | 70 | 0 | 0 | 0 |
| 42 | 45 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 13 |
| 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 45 | 0 | 0 |
G:=sub<GL(5,GF(73))| [1,0,0,0,0,0,1,0,0,0,0,0,27,0,0,0,0,0,72,0,0,0,0,0,27],[1,0,0,0,0,0,1,0,0,0,0,0,72,0,0,0,0,0,27,0,0,0,0,0,27],[28,3,0,0,0,70,31,0,0,0,0,0,0,0,28,0,0,13,0,0,0,0,0,72,0],[28,42,0,0,0,70,45,0,0,0,0,0,0,0,45,0,0,0,72,0,0,0,13,0,0] >;
C42⋊D9 in GAP, Magma, Sage, TeX
C_4^2\rtimes D_9
% in TeX
G:=Group("C4^2:D9"); // GroupNames label
G:=SmallGroup(288,67);
// by ID
G=gap.SmallGroup(288,67);
# by ID
G:=PCGroup([7,-2,-3,-3,-2,2,-2,2,141,92,254,1011,514,360,634,3476,102,9077,3036,5305]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^4=c^9=d^2=1,a*b=b*a,c*a*c^-1=d*b*d=a^-1*b^-1,a*d=d*a,c*b*c^-1=a,d*c*d=c^-1>;
// generators/relations
Export
Subgroup lattice of C42⋊D9 in TeX
Character table of C42⋊D9 in TeX