non-abelian, soluble, monomial
Aliases: F9⋊C4, C2.3AΓL1(𝔽9), C32⋊C4.Q8, C3⋊S3.SD16, C32⋊(C4.Q8), (C2×F9).3C2, (C3×C6).3SD16, C2.PSU3(𝔽2).1C2, C3⋊S3.(C4⋊C4), (C2×C3⋊S3).3D4, C32⋊C4.3(C2×C4), C3⋊S3.Q8.3C2, (C2×C32⋊C4).3C22, SmallGroup(288,843)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for F9⋊C4
G = < a,b,c,d | a3=b3=c8=d4=1, cac-1=ab=ba, dad-1=a-1b, cbc-1=a, bd=db, dcd-1=c3 >
Character table of F9⋊C4
| class | 1 | 2A | 2B | 2C | 3 | 4A | 4B | 4C | 4D | 4E | 4F | 6 | 8A | 8B | 8C | 8D | 12A | 12B | |
| size | 1 | 1 | 9 | 9 | 8 | 12 | 12 | 18 | 18 | 36 | 36 | 8 | 18 | 18 | 18 | 18 | 24 | 24 | |
| 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 | 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 | linear of order 2 |
| ρ5 | 1 | -1 | -1 | 1 | 1 | i | -i | 1 | -1 | -i | i | -1 | -1 | 1 | -1 | 1 | i | -i | linear of order 4 |
| ρ6 | 1 | -1 | -1 | 1 | 1 | -i | i | 1 | -1 | i | -i | -1 | -1 | 1 | -1 | 1 | -i | i | linear of order 4 |
| ρ7 | 1 | -1 | -1 | 1 | 1 | -i | i | 1 | -1 | -i | i | -1 | 1 | -1 | 1 | -1 | -i | i | linear of order 4 |
| ρ8 | 1 | -1 | -1 | 1 | 1 | i | -i | 1 | -1 | i | -i | -1 | 1 | -1 | 1 | -1 | i | -i | linear of order 4 |
| ρ9 | 2 | 2 | 2 | 2 | 2 | 0 | 0 | -2 | -2 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from D4 |
| ρ10 | 2 | -2 | -2 | 2 | 2 | 0 | 0 | -2 | 2 | 0 | 0 | -2 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic lifted from Q8, Schur index 2 |
| ρ11 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | √-2 | √-2 | -√-2 | -√-2 | 0 | 0 | complex lifted from SD16 |
| ρ12 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | -√-2 | √-2 | √-2 | -√-2 | 0 | 0 | complex lifted from SD16 |
| ρ13 | 2 | 2 | -2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 2 | √-2 | -√-2 | -√-2 | √-2 | 0 | 0 | complex lifted from SD16 |
| ρ14 | 2 | -2 | 2 | -2 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | -2 | -√-2 | -√-2 | √-2 | √-2 | 0 | 0 | complex lifted from SD16 |
| ρ15 | 8 | 8 | 0 | 0 | -1 | -2 | -2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from AΓL1(𝔽9) |
| ρ16 | 8 | 8 | 0 | 0 | -1 | 2 | 2 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from AΓL1(𝔽9) |
| ρ17 | 8 | -8 | 0 | 0 | -1 | 2i | -2i | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -i | i | complex faithful |
| ρ18 | 8 | -8 | 0 | 0 | -1 | -2i | 2i | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | i | -i | complex faithful |
►On 36 points
Generators in S36
(1 15 19)(2 31 35)(3 22 26)(4 12 8)(5 11 10)(6 7 9)(13 16 14)(17 18 20)(21 28 23)(24 25 27)(29 32 30)(33 34 36)
(1 16 20)(2 32 36)(3 23 27)(4 5 9)(6 12 11)(7 8 10)(13 18 19)(14 17 15)(21 24 22)(25 26 28)(29 34 35)(30 33 31)
(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 3 4 2)(5 32 16 23)(6 35 17 26)(7 30 18 21)(8 33 19 24)(9 36 20 27)(10 31 13 22)(11 34 14 25)(12 29 15 28)
G:=sub<Sym(36)| (1,15,19)(2,31,35)(3,22,26)(4,12,8)(5,11,10)(6,7,9)(13,16,14)(17,18,20)(21,28,23)(24,25,27)(29,32,30)(33,34,36), (1,16,20)(2,32,36)(3,23,27)(4,5,9)(6,12,11)(7,8,10)(13,18,19)(14,17,15)(21,24,22)(25,26,28)(29,34,35)(30,33,31), (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), (1,3,4,2)(5,32,16,23)(6,35,17,26)(7,30,18,21)(8,33,19,24)(9,36,20,27)(10,31,13,22)(11,34,14,25)(12,29,15,28)>;
G:=Group( (1,15,19)(2,31,35)(3,22,26)(4,12,8)(5,11,10)(6,7,9)(13,16,14)(17,18,20)(21,28,23)(24,25,27)(29,32,30)(33,34,36), (1,16,20)(2,32,36)(3,23,27)(4,5,9)(6,12,11)(7,8,10)(13,18,19)(14,17,15)(21,24,22)(25,26,28)(29,34,35)(30,33,31), (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), (1,3,4,2)(5,32,16,23)(6,35,17,26)(7,30,18,21)(8,33,19,24)(9,36,20,27)(10,31,13,22)(11,34,14,25)(12,29,15,28) );
G=PermutationGroup([[(1,15,19),(2,31,35),(3,22,26),(4,12,8),(5,11,10),(6,7,9),(13,16,14),(17,18,20),(21,28,23),(24,25,27),(29,32,30),(33,34,36)], [(1,16,20),(2,32,36),(3,23,27),(4,5,9),(6,12,11),(7,8,10),(13,18,19),(14,17,15),(21,24,22),(25,26,28),(29,34,35),(30,33,31)], [(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)], [(1,3,4,2),(5,32,16,23),(6,35,17,26),(7,30,18,21),(8,33,19,24),(9,36,20,27),(10,31,13,22),(11,34,14,25),(12,29,15,28)]])
Matrix representation of F9⋊C4 ►in GL8(𝔽73)
| 0 | 0 | 1 | 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 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 72 | 72 | 72 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 72 | 72 | 72 | 72 | 72 | 72 | 72 | 72 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 0 | 1 | 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 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |
| 0 | 27 | 0 | 0 | 0 | 0 | 0 | 0 |
| 27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 27 | 0 | 0 | 0 |
| 0 | 0 | 0 | 27 | 0 | 0 | 0 | 0 |
| 0 | 0 | 27 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 27 | 0 |
| 0 | 0 | 0 | 0 | 0 | 27 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27 |
G:=sub<GL(8,GF(73))| [0,0,0,0,0,0,72,1,0,0,0,0,0,1,72,0,1,0,0,0,0,0,72,0,0,1,0,0,0,0,72,0,0,0,0,0,0,0,72,0,0,0,0,1,0,0,72,0,0,0,0,0,1,0,72,0,0,0,1,0,0,0,72,0],[0,0,0,72,1,0,0,0,0,0,1,72,0,0,0,0,0,0,0,72,0,0,1,0,0,0,0,72,0,0,0,1,0,0,0,72,0,1,0,0,1,0,0,72,0,0,0,0,0,1,0,72,0,0,0,0,0,0,0,72,0,0,0,0],[0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0],[0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,0,0,27,0,0,0,0,0,0,27,0,0,0,0,0,0,0,0,0,27] >;
F9⋊C4 in GAP, Magma, Sage, TeX
F_9\rtimes C_4
% in TeX
G:=Group("F9:C4"); // GroupNames label
G:=SmallGroup(288,843);
// by ID
G=gap.SmallGroup(288,843);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,64,219,100,4037,4716,2371,201,10982,4717,3156,622]);
// Polycyclic
G:=Group<a,b,c,d|a^3=b^3=c^8=d^4=1,c*a*c^-1=a*b=b*a,d*a*d^-1=a^-1*b,c*b*c^-1=a,b*d=d*b,d*c*d^-1=c^3>;
// generators/relations
Export
Subgroup lattice of F9⋊C4 in TeX
Character table of F9⋊C4 in TeX