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 46 25 38)(18 39 26 47)(19 48 27 40)(20 41 28 33)(21 34 29 42)(22 43 30 35)(23 36 31 44)(24 45 32 37)
(2 26 43)(3 27 44)(4 45 28)(6 47 30)(7 48 31)(8 32 33)(10 18 35)(11 19 36)(12 37 20)(14 39 22)(15 40 23)(16 24 41)
(1 25 42)(3 27 44)(4 28 45)(5 46 29)(7 48 31)(8 33 32)(9 17 34)(11 19 36)(12 20 37)(13 38 21)(15 40 23)(16 41 24)
(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)(17,46,25,38)(18,39,26,47)(19,48,27,40)(20,41,28,33)(21,34,29,42)(22,43,30,35)(23,36,31,44)(24,45,32,37), (2,26,43)(3,27,44)(4,45,28)(6,47,30)(7,48,31)(8,32,33)(10,18,35)(11,19,36)(12,37,20)(14,39,22)(15,40,23)(16,24,41), (1,25,42)(3,27,44)(4,28,45)(5,46,29)(7,48,31)(8,33,32)(9,17,34)(11,19,36)(12,20,37)(13,38,21)(15,40,23)(16,41,24), (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)(17,46,25,38)(18,39,26,47)(19,48,27,40)(20,41,28,33)(21,34,29,42)(22,43,30,35)(23,36,31,44)(24,45,32,37), (2,26,43)(3,27,44)(4,45,28)(6,47,30)(7,48,31)(8,32,33)(10,18,35)(11,19,36)(12,37,20)(14,39,22)(15,40,23)(16,24,41), (1,25,42)(3,27,44)(4,28,45)(5,46,29)(7,48,31)(8,33,32)(9,17,34)(11,19,36)(12,20,37)(13,38,21)(15,40,23)(16,41,24), (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),(17,46,25,38),(18,39,26,47),(19,48,27,40),(20,41,28,33),(21,34,29,42),(22,43,30,35),(23,36,31,44),(24,45,32,37)], [(2,26,43),(3,27,44),(4,45,28),(6,47,30),(7,48,31),(8,32,33),(10,18,35),(11,19,36),(12,37,20),(14,39,22),(15,40,23),(16,24,41)], [(1,25,42),(3,27,44),(4,28,45),(5,46,29),(7,48,31),(8,33,32),(9,17,34),(11,19,36),(12,20,37),(13,38,21),(15,40,23),(16,41,24)], [(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