Aliases: C9⋊A4, C22⋊13- 1+2, (C3×A4).C3, (C2×C18)⋊2C3, C3.A4⋊1C3, C3.3(C3×A4), (C2×C6).2C32, SmallGroup(108,19)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C9⋊A4
G = < a,b,c,d | a9=b2=c2=d3=1, ab=ba, ac=ca, dad-1=a7, dbd-1=bc=cb, dcd-1=b >
Character table of C9⋊A4
| class | 1 | 2 | 3A | 3B | 3C | 3D | 6A | 6B | 9A | 9B | 9C | 9D | 9E | 9F | 18A | 18B | 18C | 18D | 18E | 18F | |
| size | 1 | 3 | 1 | 1 | 12 | 12 | 3 | 3 | 3 | 3 | 12 | 12 | 12 | 12 | 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 | trivial |
| ρ2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ3 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ4 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | 1 | 1 | 1 | 1 | linear of order 3 |
| ρ5 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ6 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ3 | ζ32 | ζ32 | ζ3 | 1 | 1 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ7 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ32 | ζ3 | linear of order 3 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | ζ3 | ζ32 | ζ3 | ζ32 | ζ3 | ζ32 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ9 | 1 | 1 | 1 | 1 | ζ32 | ζ3 | 1 | 1 | ζ3 | ζ32 | 1 | 1 | ζ32 | ζ3 | ζ32 | ζ3 | ζ3 | ζ32 | ζ3 | ζ32 | linear of order 3 |
| ρ10 | 3 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | 3 | 3 | 0 | 0 | 0 | 0 | -1 | -1 | -1 | -1 | -1 | -1 | orthogonal lifted from A4 |
| ρ11 | 3 | 3 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
| ρ12 | 3 | 3 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from 3- 1+2 |
| ρ13 | 3 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | 0 | 0 | ζ6 | ζ65 | ζ65 | ζ6 | ζ65 | ζ6 | complex lifted from C3×A4 |
| ρ14 | 3 | -1 | 3 | 3 | 0 | 0 | -1 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | 0 | 0 | ζ65 | ζ6 | ζ6 | ζ65 | ζ6 | ζ65 | complex lifted from C3×A4 |
| ρ15 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ98 | 2ζ9 | 2ζ97 | 2ζ92 | 2ζ94 | 2ζ95 | complex faithful |
| ρ16 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ95 | 2ζ94 | 2ζ9 | 2ζ98 | 2ζ97 | 2ζ92 | complex faithful |
| ρ17 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ94 | 2ζ95 | 2ζ98 | 2ζ9 | 2ζ92 | 2ζ97 | complex faithful |
| ρ18 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ97 | 2ζ92 | 2ζ95 | 2ζ94 | 2ζ98 | 2ζ9 | complex faithful |
| ρ19 | 3 | -1 | -3-3√-3/2 | -3+3√-3/2 | 0 | 0 | ζ6 | ζ65 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ92 | 2ζ97 | 2ζ94 | 2ζ95 | 2ζ9 | 2ζ98 | complex faithful |
| ρ20 | 3 | -1 | -3+3√-3/2 | -3-3√-3/2 | 0 | 0 | ζ65 | ζ6 | 0 | 0 | 0 | 0 | 0 | 0 | 2ζ9 | 2ζ98 | 2ζ92 | 2ζ97 | 2ζ95 | 2ζ94 | complex faithful |
►On 36 points
Generators in S36
(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 15)(2 16)(3 17)(4 18)(5 10)(6 11)(7 12)(8 13)(9 14)(19 35)(20 36)(21 28)(22 29)(23 30)(24 31)(25 32)(26 33)(27 34)
(1 25)(2 26)(3 27)(4 19)(5 20)(6 21)(7 22)(8 23)(9 24)(10 36)(11 28)(12 29)(13 30)(14 31)(15 32)(16 33)(17 34)(18 35)
(2 5 8)(3 9 6)(10 23 33)(11 27 31)(12 22 29)(13 26 36)(14 21 34)(15 25 32)(16 20 30)(17 24 28)(18 19 35)
G:=sub<Sym(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), (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (2,5,8)(3,9,6)(10,23,33)(11,27,31)(12,22,29)(13,26,36)(14,21,34)(15,25,32)(16,20,30)(17,24,28)(18,19,35)>;
G:=Group( (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), (1,15)(2,16)(3,17)(4,18)(5,10)(6,11)(7,12)(8,13)(9,14)(19,35)(20,36)(21,28)(22,29)(23,30)(24,31)(25,32)(26,33)(27,34), (1,25)(2,26)(3,27)(4,19)(5,20)(6,21)(7,22)(8,23)(9,24)(10,36)(11,28)(12,29)(13,30)(14,31)(15,32)(16,33)(17,34)(18,35), (2,5,8)(3,9,6)(10,23,33)(11,27,31)(12,22,29)(13,26,36)(14,21,34)(15,25,32)(16,20,30)(17,24,28)(18,19,35) );
G=PermutationGroup([[(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)], [(1,15),(2,16),(3,17),(4,18),(5,10),(6,11),(7,12),(8,13),(9,14),(19,35),(20,36),(21,28),(22,29),(23,30),(24,31),(25,32),(26,33),(27,34)], [(1,25),(2,26),(3,27),(4,19),(5,20),(6,21),(7,22),(8,23),(9,24),(10,36),(11,28),(12,29),(13,30),(14,31),(15,32),(16,33),(17,34),(18,35)], [(2,5,8),(3,9,6),(10,23,33),(11,27,31),(12,22,29),(13,26,36),(14,21,34),(15,25,32),(16,20,30),(17,24,28),(18,19,35)]])
C9⋊A4 is a maximal subgroup of
D9⋊A4 C62.25C32 A4×3- 1+2 C62.9C32 C42⋊3- 1+2 C24⋊3- 1+2 C24⋊23- 1+2 C24⋊43- 1+2
C9⋊A4 is a maximal quotient of
C18.A4 C62.11C32 C62.12C32 C62.16C32 C42⋊3- 1+2 C24⋊3- 1+2 C24⋊23- 1+2 C24⋊43- 1+2
Matrix representation of C9⋊A4 ►in GL3(𝔽19) generated by
| 11 | 4 | 10 |
| 13 | 1 | 9 |
| 6 | 15 | 7 |
| 0 | 0 | 1 |
| 18 | 18 | 18 |
| 1 | 0 | 0 |
| 0 | 1 | 0 |
| 1 | 0 | 0 |
| 18 | 18 | 18 |
| 1 | 0 | 0 |
| 0 | 0 | 1 |
| 18 | 18 | 18 |
G:=sub<GL(3,GF(19))| [11,13,6,4,1,15,10,9,7],[0,18,1,0,18,0,1,18,0],[0,1,18,1,0,18,0,0,18],[1,0,18,0,0,18,0,1,18] >;
C9⋊A4 in GAP, Magma, Sage, TeX
C_9\rtimes A_4
% in TeX
G:=Group("C9:A4"); // GroupNames label
G:=SmallGroup(108,19);
// by ID
G=gap.SmallGroup(108,19);
# by ID
G:=PCGroup([5,-3,-3,-3,-2,2,121,36,1083,2029]);
// Polycyclic
G:=Group<a,b,c,d|a^9=b^2=c^2=d^3=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^7,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C9⋊A4 in TeX
Character table of C9⋊A4 in TeX