Aliases: C22.F9, C62.2C8, C32⋊2M5(2), C2.F9⋊2C2, C2.6(C2×F9), C3⋊Dic3.2C8, C32⋊2C8.6C4, C32⋊2C8.6C22, (C3×C6).6(C2×C8), (C2×C3⋊Dic3).5C4, C3⋊Dic3.4(C2×C4), (C2×C32⋊2C8).10C2, SmallGroup(288,866)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C22.F9
G = < a,b,c,d,e | a2=b2=c3=d3=1, e8=b, eae-1=ab=ba, ac=ca, ad=da, bc=cb, bd=db, be=eb, ece-1=cd=dc, ede-1=c >
Character table of C22.F9
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 8A | 8B | 8C | 8D | 8E | 8F | 16A | 16B | 16C | 16D | 16E | 16F | 16G | 16H | |
| size | 1 | 1 | 2 | 8 | 9 | 9 | 18 | 8 | 8 | 8 | 9 | 9 | 9 | 9 | 18 | 18 | 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 | 1 | -1 | -i | -i | i | i | i | -i | ζ85 | ζ83 | ζ85 | ζ8 | ζ83 | ζ87 | ζ8 | ζ87 | linear of order 8 |
| ρ10 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -i | i | ζ85 | ζ87 | ζ8 | ζ85 | ζ83 | ζ87 | ζ8 | ζ83 | linear of order 8 |
| ρ11 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | i | -i | ζ83 | ζ8 | ζ87 | ζ83 | ζ85 | ζ8 | ζ87 | ζ85 | linear of order 8 |
| ρ12 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | -i | -i | i | i | -i | i | ζ8 | ζ83 | ζ85 | ζ8 | ζ87 | ζ83 | ζ85 | ζ87 | linear of order 8 |
| ρ13 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | i | ζ83 | ζ85 | ζ83 | ζ87 | ζ85 | ζ8 | ζ87 | ζ8 | linear of order 8 |
| ρ14 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | -i | -i | i | i | i | -i | ζ8 | ζ87 | ζ8 | ζ85 | ζ87 | ζ83 | ζ85 | ζ83 | linear of order 8 |
| ρ15 | 1 | 1 | -1 | 1 | -1 | -1 | 1 | -1 | 1 | -1 | i | i | -i | -i | -i | i | ζ87 | ζ8 | ζ87 | ζ83 | ζ8 | ζ85 | ζ83 | ζ85 | linear of order 8 |
| ρ16 | 1 | 1 | 1 | 1 | -1 | -1 | -1 | 1 | 1 | 1 | i | i | -i | -i | i | -i | ζ87 | ζ85 | ζ83 | ζ87 | ζ8 | ζ85 | ζ83 | ζ8 | linear of order 8 |
| ρ17 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | 0 | 2ζ8 | 2ζ85 | 2ζ83 | 2ζ87 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ18 | 2 | -2 | 0 | 2 | 2i | -2i | 0 | 0 | -2 | 0 | 2ζ85 | 2ζ8 | 2ζ87 | 2ζ83 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ19 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | 0 | 2ζ83 | 2ζ87 | 2ζ8 | 2ζ85 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ20 | 2 | -2 | 0 | 2 | -2i | 2i | 0 | 0 | -2 | 0 | 2ζ87 | 2ζ83 | 2ζ85 | 2ζ8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | complex lifted from M5(2) |
| ρ21 | 8 | 8 | 8 | -1 | 0 | 0 | 0 | -1 | -1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from F9 |
| ρ22 | 8 | 8 | -8 | -1 | 0 | 0 | 0 | 1 | -1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | orthogonal lifted from C2×F9 |
| ρ23 | 8 | -8 | 0 | -1 | 0 | 0 | 0 | 3 | 1 | -3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
| ρ24 | 8 | -8 | 0 | -1 | 0 | 0 | 0 | -3 | 1 | 3 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | symplectic faithful, Schur index 2 |
►On 48 points
Generators in S48
(2 10)(4 12)(6 14)(8 16)(18 26)(20 28)(22 30)(24 32)(33 41)(35 43)(37 45)(39 47)
(1 9)(2 10)(3 11)(4 12)(5 13)(6 14)(7 15)(8 16)(17 25)(18 26)(19 27)(20 28)(21 29)(22 30)(23 31)(24 32)(33 41)(34 42)(35 43)(36 44)(37 45)(38 46)(39 47)(40 48)
(2 30 45)(3 31 46)(4 47 32)(6 33 18)(7 34 19)(8 20 35)(10 22 37)(11 23 38)(12 39 24)(14 41 26)(15 42 27)(16 28 43)
(1 29 44)(3 31 46)(4 32 47)(5 48 17)(7 34 19)(8 35 20)(9 21 36)(11 23 38)(12 24 39)(13 40 25)(15 42 27)(16 43 28)
(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)| (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,30,45)(3,31,46)(4,47,32)(6,33,18)(7,34,19)(8,20,35)(10,22,37)(11,23,38)(12,39,24)(14,41,26)(15,42,27)(16,28,43), (1,29,44)(3,31,46)(4,32,47)(5,48,17)(7,34,19)(8,35,20)(9,21,36)(11,23,38)(12,24,39)(13,40,25)(15,42,27)(16,43,28), (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( (2,10)(4,12)(6,14)(8,16)(18,26)(20,28)(22,30)(24,32)(33,41)(35,43)(37,45)(39,47), (1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)(17,25)(18,26)(19,27)(20,28)(21,29)(22,30)(23,31)(24,32)(33,41)(34,42)(35,43)(36,44)(37,45)(38,46)(39,47)(40,48), (2,30,45)(3,31,46)(4,47,32)(6,33,18)(7,34,19)(8,20,35)(10,22,37)(11,23,38)(12,39,24)(14,41,26)(15,42,27)(16,28,43), (1,29,44)(3,31,46)(4,32,47)(5,48,17)(7,34,19)(8,35,20)(9,21,36)(11,23,38)(12,24,39)(13,40,25)(15,42,27)(16,43,28), (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([[(2,10),(4,12),(6,14),(8,16),(18,26),(20,28),(22,30),(24,32),(33,41),(35,43),(37,45),(39,47)], [(1,9),(2,10),(3,11),(4,12),(5,13),(6,14),(7,15),(8,16),(17,25),(18,26),(19,27),(20,28),(21,29),(22,30),(23,31),(24,32),(33,41),(34,42),(35,43),(36,44),(37,45),(38,46),(39,47),(40,48)], [(2,30,45),(3,31,46),(4,47,32),(6,33,18),(7,34,19),(8,20,35),(10,22,37),(11,23,38),(12,39,24),(14,41,26),(15,42,27),(16,28,43)], [(1,29,44),(3,31,46),(4,32,47),(5,48,17),(7,34,19),(8,35,20),(9,21,36),(11,23,38),(12,24,39),(13,40,25),(15,42,27),(16,43,28)], [(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 C22.F9 ►in GL10(𝔽97)
| 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 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 | 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 |
| 96 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 0 | 96 | 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 | 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 |
| 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 | 50 | 50 | 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 | 47 | 0 | 1 |
| 0 | 0 | 0 | 0 | 0 | 0 | 50 | 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 | 47 | 0 | 1 | 0 | 0 | 0 | 0 |
| 0 | 0 | 50 | 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 | 47 | 0 | 1 | 0 |
| 0 | 0 | 0 | 0 | 0 | 0 | 47 | 0 | 0 | 1 |
| 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 50 | 0 | 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 | 50 | 50 | 95 | 96 | 0 | 0 | 0 | 0 |
| 0 | 0 | 40 | 8 | 47 | 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 8 | 27 | 47 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(10,GF(97))| [1,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[96,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,50,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,50,0,0,0,0,0,0,96,96,47,0,0,0,0,0,0,0,0,0,0,96,0,0,0,0,0,0,0,0,1,96,0,0,0,0,0,0,0,0,0,0,96,96,47,47,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[0,50,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,50,40,8,0,0,0,0,0,0,0,50,8,27,0,0,0,0,0,0,96,95,47,47,0,0,0,0,0,0,1,96,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C22.F9 in GAP, Magma, Sage, TeX
C_2^2.F_9
% in TeX
G:=Group("C2^2.F9"); // GroupNames label
G:=SmallGroup(288,866);
// by ID
G=gap.SmallGroup(288,866);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,253,58,80,4037,2371,362,10982,3156,1203]);
// Polycyclic
G:=Group<a,b,c,d,e|a^2=b^2=c^3=d^3=1,e^8=b,e*a*e^-1=a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,b*e=e*b,e*c*e^-1=c*d=d*c,e*d*e^-1=c>;
// generators/relations
Export
Subgroup lattice of C22.F9 in TeX
Character table of C22.F9 in TeX