metabelian, soluble, monomial, A-group
Aliases: C42⋊C9, (C4×C12).C3, (C2×C6).1A4, C3.(C42⋊C3), C22.(C3.A4), SmallGroup(144,3)
Series: Derived ►Chief ►Lower central ►Upper central
| C42 — C42⋊C9 |
Generators and relations for C42⋊C9
G = < a,b,c | a4=b4=c9=1, ab=ba, cac-1=ab-1, cbc-1=a-1b2 >
Character table of C42⋊C9
| class | 1 | 2 | 3A | 3B | 4A | 4B | 4C | 4D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 12A | 12B | 12C | 12D | 12E | 12F | 12G | 12H | |
| size | 1 | 3 | 1 | 1 | 3 | 3 | 3 | 3 | 3 | 3 | 16 | 16 | 16 | 16 | 16 | 16 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |
| ρ1 | 1 | 1 | 1 | 1 | 1 | 1 | 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 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ9 | ζ95 | ζ92 | ζ97 | ζ94 | ζ98 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ5 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ98 | ζ94 | ζ97 | ζ92 | ζ95 | ζ9 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ6 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ92 | ζ9 | ζ94 | ζ95 | ζ98 | ζ97 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ7 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ94 | ζ92 | ζ98 | ζ9 | ζ97 | ζ95 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ8 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ95 | ζ97 | ζ9 | ζ98 | ζ92 | ζ94 | ζ32 | ζ32 | ζ3 | ζ3 | ζ3 | ζ3 | ζ32 | ζ32 | linear of order 9 |
| ρ9 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ97 | ζ98 | ζ95 | ζ94 | ζ9 | ζ92 | ζ3 | ζ3 | ζ32 | ζ32 | ζ32 | ζ32 | ζ3 | ζ3 | linear of order 9 |
| ρ10 | 3 | 3 | 3 | 3 | -1 | -1 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | -1 | -1 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ65 | ζ65 | ζ6 | ζ6 | ζ6 | ζ6 | ζ65 | ζ65 | complex lifted from C3.A4 |
| ρ12 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | -1 | -1 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | ζ6 | ζ6 | ζ65 | ζ65 | ζ65 | ζ65 | ζ6 | ζ6 | complex lifted from C3.A4 |
| ρ13 | 3 | -1 | 3 | 3 | -1-2i | 1 | 1 | -1+2i | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1-2i | 1 | -1-2i | 1 | 1 | -1+2i | 1 | -1+2i | complex lifted from C42⋊C3 |
| ρ14 | 3 | -1 | 3 | 3 | 1 | -1+2i | -1-2i | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1+2i | 1 | -1+2i | -1-2i | 1 | -1-2i | 1 | complex lifted from C42⋊C3 |
| ρ15 | 3 | -1 | 3 | 3 | 1 | -1-2i | -1+2i | 1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | -1-2i | 1 | -1-2i | -1+2i | 1 | -1+2i | 1 | complex lifted from C42⋊C3 |
| ρ16 | 3 | -1 | 3 | 3 | -1+2i | 1 | 1 | -1-2i | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1+2i | 1 | -1+2i | 1 | 1 | -1-2i | 1 | -1-2i | complex lifted from C42⋊C3 |
| ρ17 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -1-2i | 1 | 1 | -1+2i | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ32-ζ32 | ζ32 | 2ζ43ζ3-ζ3 | ζ3 | ζ3 | 2ζ4ζ3-ζ3 | ζ32 | 2ζ4ζ32-ζ32 | complex faithful |
| ρ18 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 1 | -1+2i | -1-2i | 1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | 2ζ4ζ32-ζ32 | ζ3 | 2ζ4ζ3-ζ3 | 2ζ43ζ3-ζ3 | ζ3 | 2ζ43ζ32-ζ32 | ζ32 | complex faithful |
| ρ19 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 1 | -1-2i | -1+2i | 1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | 2ζ43ζ3-ζ3 | ζ32 | 2ζ43ζ32-ζ32 | 2ζ4ζ32-ζ32 | ζ32 | 2ζ4ζ3-ζ3 | ζ3 | complex faithful |
| ρ20 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -1-2i | 1 | 1 | -1+2i | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ43ζ3-ζ3 | ζ3 | 2ζ43ζ32-ζ32 | ζ32 | ζ32 | 2ζ4ζ32-ζ32 | ζ3 | 2ζ4ζ3-ζ3 | complex faithful |
| ρ21 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 1 | -1-2i | -1+2i | 1 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | ζ32 | 2ζ43ζ32-ζ32 | ζ3 | 2ζ43ζ3-ζ3 | 2ζ4ζ3-ζ3 | ζ3 | 2ζ4ζ32-ζ32 | ζ32 | complex faithful |
| ρ22 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | -1+2i | 1 | 1 | -1-2i | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ3-ζ3 | ζ3 | 2ζ4ζ32-ζ32 | ζ32 | ζ32 | 2ζ43ζ32-ζ32 | ζ3 | 2ζ43ζ3-ζ3 | complex faithful |
| ρ23 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 1 | -1+2i | -1-2i | 1 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | ζ3 | 2ζ4ζ3-ζ3 | ζ32 | 2ζ4ζ32-ζ32 | 2ζ43ζ32-ζ32 | ζ32 | 2ζ43ζ3-ζ3 | ζ3 | complex faithful |
| ρ24 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | -1+2i | 1 | 1 | -1-2i | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ4ζ32-ζ32 | ζ32 | 2ζ4ζ3-ζ3 | ζ3 | ζ3 | 2ζ43ζ3-ζ3 | ζ32 | 2ζ43ζ32-ζ32 | complex faithful |
►On 36 points
Generators in S36
(2 24 18 33)(3 34 10 25)(5 27 12 36)(6 28 13 19)(8 21 15 30)(9 31 16 22)
(1 32 17 23)(2 18)(3 34 10 25)(4 35 11 26)(5 12)(6 28 13 19)(7 29 14 20)(8 15)(9 31 16 22)(21 30)(24 33)(27 36)
(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)| (2,24,18,33)(3,34,10,25)(5,27,12,36)(6,28,13,19)(8,21,15,30)(9,31,16,22), (1,32,17,23)(2,18)(3,34,10,25)(4,35,11,26)(5,12)(6,28,13,19)(7,29,14,20)(8,15)(9,31,16,22)(21,30)(24,33)(27,36), (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( (2,24,18,33)(3,34,10,25)(5,27,12,36)(6,28,13,19)(8,21,15,30)(9,31,16,22), (1,32,17,23)(2,18)(3,34,10,25)(4,35,11,26)(5,12)(6,28,13,19)(7,29,14,20)(8,15)(9,31,16,22)(21,30)(24,33)(27,36), (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([[(2,24,18,33),(3,34,10,25),(5,27,12,36),(6,28,13,19),(8,21,15,30),(9,31,16,22)], [(1,32,17,23),(2,18),(3,34,10,25),(4,35,11,26),(5,12),(6,28,13,19),(7,29,14,20),(8,15),(9,31,16,22),(21,30),(24,33),(27,36)], [(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)]])
C42⋊C9 is a maximal subgroup of
C42⋊D9 C42⋊C18 C42⋊2C18 C9×C42⋊C3 C42⋊3- 1+2 C122.C3
C42⋊C9 is a maximal quotient of C2.(C42⋊C9) C42⋊C27
Matrix representation of C42⋊C9 ►in GL3(𝔽13) generated by
| 5 | 0 | 0 |
| 0 | 1 | 0 |
| 0 | 0 | 8 |
| 5 | 0 | 0 |
| 0 | 5 | 0 |
| 0 | 0 | 12 |
| 0 | 0 | 9 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
G:=sub<GL(3,GF(13))| [5,0,0,0,1,0,0,0,8],[5,0,0,0,5,0,0,0,12],[0,1,0,0,0,1,9,0,0] >;
C42⋊C9 in GAP, Magma, Sage, TeX
C_4^2\rtimes C_9
% in TeX
G:=Group("C4^2:C9"); // GroupNames label
G:=SmallGroup(144,3);
// by ID
G=gap.SmallGroup(144,3);
# by ID
G:=PCGroup([6,-3,-3,-2,2,-2,2,18,326,230,2379,69,2164,3893]);
// Polycyclic
G:=Group<a,b,c|a^4=b^4=c^9=1,a*b=b*a,c*a*c^-1=a*b^-1,c*b*c^-1=a^-1*b^2>;
// generators/relations
Export
Subgroup lattice of C42⋊C9 in TeX
Character table of C42⋊C9 in TeX