Aliases: C4⋊F9, (C3×C12)⋊2C8, C2.5(C2×F9), C32⋊1(C4⋊C8), C3⋊Dic3⋊2C8, (C2×F9).2C2, C32⋊C4.4D4, C32⋊C4.2Q8, C3⋊S3.1M4(2), (C4×C3⋊S3).4C4, (C3×C6).4(C2×C8), C3⋊S3.2(C4⋊C4), (C4×C32⋊C4).7C2, (C2×C32⋊C4).5C4, (C2×C32⋊C4).10C22, (C2×C3⋊S3).2(C2×C4), SmallGroup(288,864)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4⋊F9
G = < a,b,c,d | a4=b3=c3=d8=1, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >
Character table of C4⋊F9
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 4G | 4H | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 8G | 8H | 12A | 12B | |
| size | 1 | 1 | 9 | 9 | 8 | 2 | 9 | 9 | 9 | 9 | 18 | 18 | 18 | 8 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 8 | 8 | |
| 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 | -1 | -1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -1 | -1 | linear of order 2 |
| ρ3 | 1 | 1 | 1 | 1 | 1 | -1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | linear of order 2 |
| ρ4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | linear of order 2 |
| ρ5 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | i | -i | i | i | -i | -i | i | -i | 1 | 1 | linear of order 4 |
| ρ6 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | -i | -i | i | i | -1 | -1 | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | i | i | -i | -i | -1 | -1 | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | -1 | -1 | 1 | -i | i | -i | -i | i | i | -i | i | 1 | 1 | linear of order 4 |
| ρ9 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -i | i | 1 | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | 1 | 1 | linear of order 8 |
| ρ10 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | -i | i | 1 | -i | i | 1 | ζ83 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | ζ8 | -1 | -1 | linear of order 8 |
| ρ11 | 1 | 1 | -1 | -1 | 1 | -1 | -i | i | -i | i | 1 | -i | i | 1 | ζ87 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | ζ85 | -1 | -1 | linear of order 8 |
| ρ12 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | i | -i | 1 | i | -i | 1 | ζ85 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | ζ87 | -1 | -1 | linear of order 8 |
| ρ13 | 1 | 1 | -1 | -1 | 1 | -1 | i | -i | i | -i | 1 | i | -i | 1 | ζ8 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | ζ83 | -1 | -1 | linear of order 8 |
| ρ14 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | i | -i | 1 | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | 1 | 1 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | -1 | 1 | 1 | -i | i | -i | i | -1 | i | -i | 1 | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | 1 | 1 | linear of order 8 |
| ρ16 | 1 | 1 | -1 | -1 | 1 | 1 | i | -i | i | -i | -1 | -i | i | 1 | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | 1 | 1 | linear of order 8 |
| ρ17 | 2 | -2 | 2 | -2 | 2 | 0 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ18 | 2 | -2 | 2 | -2 | 2 | 0 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ19 | 2 | -2 | -2 | 2 | 2 | 0 | -2i | -2i | 2i | 2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ20 | 2 | -2 | -2 | 2 | 2 | 0 | 2i | 2i | -2i | -2i | 0 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M4(2) |
| ρ21 | 8 | 8 | 0 | 0 | -1 | 8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from F9 |
| ρ22 | 8 | 8 | 0 | 0 | -1 | -8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C2×F9 |
| ρ23 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | complex faithful |
| ρ24 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | complex faithful |
►On 36 points
Generators in S36
(1 3 2 4)(5 20 35 22)(6 23 36 13)(7 14 29 24)(8 25 30 15)(9 16 31 26)(10 27 32 17)(11 18 33 28)(12 21 34 19)
(1 7 11)(2 29 33)(3 14 18)(4 24 28)(5 8 6)(9 10 12)(13 22 15)(16 27 21)(17 19 26)(20 25 23)(30 36 35)(31 32 34)
(1 8 12)(2 30 34)(3 25 21)(4 15 19)(5 10 11)(6 9 7)(13 26 24)(14 23 16)(17 28 22)(18 20 27)(29 36 31)(32 33 35)
(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)| (1,3,2,4)(5,20,35,22)(6,23,36,13)(7,14,29,24)(8,25,30,15)(9,16,31,26)(10,27,32,17)(11,18,33,28)(12,21,34,19), (1,7,11)(2,29,33)(3,14,18)(4,24,28)(5,8,6)(9,10,12)(13,22,15)(16,27,21)(17,19,26)(20,25,23)(30,36,35)(31,32,34), (1,8,12)(2,30,34)(3,25,21)(4,15,19)(5,10,11)(6,9,7)(13,26,24)(14,23,16)(17,28,22)(18,20,27)(29,36,31)(32,33,35), (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( (1,3,2,4)(5,20,35,22)(6,23,36,13)(7,14,29,24)(8,25,30,15)(9,16,31,26)(10,27,32,17)(11,18,33,28)(12,21,34,19), (1,7,11)(2,29,33)(3,14,18)(4,24,28)(5,8,6)(9,10,12)(13,22,15)(16,27,21)(17,19,26)(20,25,23)(30,36,35)(31,32,34), (1,8,12)(2,30,34)(3,25,21)(4,15,19)(5,10,11)(6,9,7)(13,26,24)(14,23,16)(17,28,22)(18,20,27)(29,36,31)(32,33,35), (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([[(1,3,2,4),(5,20,35,22),(6,23,36,13),(7,14,29,24),(8,25,30,15),(9,16,31,26),(10,27,32,17),(11,18,33,28),(12,21,34,19)], [(1,7,11),(2,29,33),(3,14,18),(4,24,28),(5,8,6),(9,10,12),(13,22,15),(16,27,21),(17,19,26),(20,25,23),(30,36,35),(31,32,34)], [(1,8,12),(2,30,34),(3,25,21),(4,15,19),(5,10,11),(6,9,7),(13,26,24),(14,23,16),(17,28,22),(18,20,27),(29,36,31),(32,33,35)], [(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)]])
Matrix representation of C4⋊F9 ►in GL10(𝔽73)
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 72 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 72 | 0 | 0 | 0 | 0 | 1 | 0 |
| 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 51 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
G:=sub<GL(10,GF(73))| [0,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72,0,0,0,0,0,0,0,0,0,0,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,72,72,72,72,72,72,72,72,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,72,72,72,72,72,72,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0],[22,0,0,0,0,0,0,0,0,0,0,51,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0] >;
C4⋊F9 in GAP, Magma, Sage, TeX
C_4\rtimes F_9
% in TeX
G:=Group("C4:F9"); // GroupNames label
G:=SmallGroup(288,864);
// by ID
G=gap.SmallGroup(288,864);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,141,64,100,4037,2371,201,10982,3156,622]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=d^8=1,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,d*b*d^-1=b*c=c*b,d*c*d^-1=b>;
// generators/relations
Export
Subgroup lattice of C4⋊F9 in TeX
Character table of C4⋊F9 in TeX