Aliases: C4.F9, C32⋊1M5(2), C2.F9⋊1C2, C2.4(C2×F9), (C3×C12).3C8, C32⋊2C8.5C4, C32⋊2C8.4C22, (C4×C3⋊S3).2C4, (C2×C3⋊S3).2C8, (C3×C6).2(C2×C8), C3⋊S3⋊3C8.9C2, C3⋊Dic3.2(C2×C4), SmallGroup(288,862)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.F9
G = < a,b,c,d | a4=b3=c3=1, d8=a2, ab=ba, ac=ca, dad-1=a-1, dbd-1=bc=cb, dcd-1=b >
Character table of C4.F9
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6 | 8A | 8B | 8C | 8D | 8E | 8F | 12A | 12B | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 18 | 8 | 2 | 9 | 9 | 8 | 9 | 9 | 9 | 9 | 18 | 18 | 8 | 8 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | 18 | |
| 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 | -1 | -1 | -i | -i | i | i | i | i | -i | -i | linear of order 4 |
| ρ6 | 1 | 1 | -1 | 1 | -1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | 1 | 1 | -1 | -1 | i | i | -i | -i | -i | -i | i | i | linear of order 4 |
| ρ7 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | -i | -i | i | i | -i | -i | i | i | linear of order 4 |
| ρ8 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | -1 | -1 | -1 | 1 | 1 | i | i | -i | -i | i | i | -i | -i | linear of order 4 |
| ρ9 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | -1 | -1 | ζ8 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | i | -1 | -1 | ζ87 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | -i | -i | i | i | i | -i | -1 | -1 | ζ85 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | i | i | -i | -i | -i | i | -1 | -1 | ζ83 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ13 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | -i | i | 1 | 1 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | linear of order 8 |
| ρ14 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | i | -i | 1 | 1 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | i | i | -i | -i | i | -i | 1 | 1 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | linear of order 8 |
| ρ16 | 1 | 1 | -1 | 1 | 1 | -1 | -1 | 1 | -i | -i | i | i | -i | i | 1 | 1 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | linear of order 8 |
| ρ17 | 2 | -2 | 0 | 2 | 0 | 2i | -2i | -2 | 2ζ8 | 2ζ85 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ18 | 2 | -2 | 0 | 2 | 0 | -2i | 2i | -2 | 2ζ87 | 2ζ83 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ19 | 2 | -2 | 0 | 2 | 0 | 2i | -2i | -2 | 2ζ85 | 2ζ8 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ20 | 2 | -2 | 0 | 2 | 0 | -2i | 2i | -2 | 2ζ83 | 2ζ87 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ21 | 8 | 8 | 0 | -1 | 8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
| ρ22 | 8 | 8 | 0 | -1 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F9 |
| ρ23 | 8 | -8 | 0 | -1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | -3i | 3i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
| ρ24 | 8 | -8 | 0 | -1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3i | -3i | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex faithful |
►On 48 points
Generators in S48
(1 13 9 5)(2 6 10 14)(3 15 11 7)(4 8 12 16)(17 45 25 37)(18 38 26 46)(19 47 27 39)(20 40 28 48)(21 33 29 41)(22 42 30 34)(23 35 31 43)(24 44 32 36)
(2 25 33)(3 26 34)(4 35 27)(6 37 29)(7 38 30)(8 31 39)(10 17 41)(11 18 42)(12 43 19)(14 45 21)(15 46 22)(16 23 47)
(1 24 48)(3 26 34)(4 27 35)(5 36 28)(7 38 30)(8 39 31)(9 32 40)(11 18 42)(12 19 43)(13 44 20)(15 46 22)(16 47 23)
(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 37 38 39 40 41 42 43 44 45 46 47 48)
G:=sub<Sym(48)| (1,13,9,5)(2,6,10,14)(3,15,11,7)(4,8,12,16)(17,45,25,37)(18,38,26,46)(19,47,27,39)(20,40,28,48)(21,33,29,41)(22,42,30,34)(23,35,31,43)(24,44,32,36), (2,25,33)(3,26,34)(4,35,27)(6,37,29)(7,38,30)(8,31,39)(10,17,41)(11,18,42)(12,43,19)(14,45,21)(15,46,22)(16,23,47), (1,24,48)(3,26,34)(4,27,35)(5,36,28)(7,38,30)(8,39,31)(9,32,40)(11,18,42)(12,19,43)(13,44,20)(15,46,22)(16,47,23), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)>;
G:=Group( (1,13,9,5)(2,6,10,14)(3,15,11,7)(4,8,12,16)(17,45,25,37)(18,38,26,46)(19,47,27,39)(20,40,28,48)(21,33,29,41)(22,42,30,34)(23,35,31,43)(24,44,32,36), (2,25,33)(3,26,34)(4,35,27)(6,37,29)(7,38,30)(8,31,39)(10,17,41)(11,18,42)(12,43,19)(14,45,21)(15,46,22)(16,23,47), (1,24,48)(3,26,34)(4,27,35)(5,36,28)(7,38,30)(8,39,31)(9,32,40)(11,18,42)(12,19,43)(13,44,20)(15,46,22)(16,47,23), (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)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48) );
G=PermutationGroup([[(1,13,9,5),(2,6,10,14),(3,15,11,7),(4,8,12,16),(17,45,25,37),(18,38,26,46),(19,47,27,39),(20,40,28,48),(21,33,29,41),(22,42,30,34),(23,35,31,43),(24,44,32,36)], [(2,25,33),(3,26,34),(4,35,27),(6,37,29),(7,38,30),(8,31,39),(10,17,41),(11,18,42),(12,43,19),(14,45,21),(15,46,22),(16,23,47)], [(1,24,48),(3,26,34),(4,27,35),(5,36,28),(7,38,30),(8,39,31),(9,32,40),(11,18,42),(12,19,43),(13,44,20),(15,46,22),(16,47,23)], [(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),(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48)]])
Matrix representation of C4.F9 ►in GL10(𝔽97)
| 22 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 75 | 75 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 |
| 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 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 96 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 96 | 96 |
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 96 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 96 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 0 | 96 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 96 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 96 | 0 | 0 | 1 |
| 96 | 95 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 74 | 1 | 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 | 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 | 96 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 1 | 1 | 95 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 33 | 96 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 33 | 32 | 96 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(10,GF(97))| [22,75,0,0,0,0,0,0,0,0,0,75,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,0,0,0,0,0,0,96,96,96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,96,96,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,74,0,0,0,0,0,0,0,0,95,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,33,33,0,0,0,0,0,0,0,1,33,32,0,0,0,0,0,0,96,95,96,96,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C4.F9 in GAP, Magma, Sage, TeX
C_4.F_9
% in TeX
G:=Group("C4.F9"); // GroupNames label
G:=SmallGroup(288,862);
// by ID
G=gap.SmallGroup(288,862);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,120,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic
G:=Group<a,b,c,d|a^4=b^3=c^3=1,d^8=a^2,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