non-abelian, soluble, monomial
Aliases: C4.1PSU3(F2), (C3xC12).2Q8, C32:2C8:4C4, C3:S3.4SD16, C32:2(C4.Q8), C2.3(C2.PSU3(F2)), (C3xC6).7(C4:C4), (C2xC3:S3).10D4, C3:S3:3C8.5C2, C4:(C32:C4).4C2, (C4xC3:S3).55C22, C3:Dic3.14(C2xC4), SmallGroup(288,393)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C4.PSU3(F2)
G = < a,b,c,d,e | a4=b3=c3=d4=1, e2=a-1d2, ab=ba, ac=ca, dad-1=a-1, ae=ea, ece-1=bc=cb, dbd-1=c-1, ebe-1=b-1c, dcd-1=b, ede-1=ad-1 >
Character table of C4.PSU3(F2)
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 | 2 | 18 | 36 | 36 | 36 | 36 | 8 | 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 | 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 | -1 | 1 | -i | -i | i | i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
ρ6 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | -i | i | i | -i | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
ρ7 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | -i | -i | i | 1 | -1 | -1 | 1 | 1 | -1 | -1 | linear of order 4 |
ρ8 | 1 | 1 | -1 | -1 | 1 | -1 | 1 | i | i | -i | -i | 1 | 1 | 1 | -1 | -1 | -1 | -1 | linear of order 4 |
ρ9 | 2 | 2 | 2 | 2 | 2 | -2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | -2 | -2 | orthogonal lifted from D4 |
ρ10 | 2 | 2 | -2 | -2 | 2 | 2 | -2 | 0 | 0 | 0 | 0 | 2 | 0 | 0 | 0 | 0 | 2 | 2 | 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 | -8 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 1 | 1 | orthogonal lifted from C2.PSU3(F2) |
ρ16 | 8 | 8 | 0 | 0 | -1 | 8 | 0 | 0 | 0 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | -1 | -1 | orthogonal lifted from PSU3(F2) |
ρ17 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | -3i | 3i | complex faithful |
ρ18 | 8 | -8 | 0 | 0 | -1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 3i | -3i | complex faithful |
(1 7 5 3)(2 8 6 4)(9 15 13 11)(10 16 14 12)(17 37 21 33)(18 38 22 34)(19 39 23 35)(20 40 24 36)(25 42 29 46)(26 43 30 47)(27 44 31 48)(28 45 32 41)
(2 20 34)(4 36 22)(6 24 38)(8 40 18)(9 43 32)(10 25 44)(11 26 45)(12 46 27)(13 47 28)(14 29 48)(15 30 41)(16 42 31)
(1 19 33)(2 20 34)(3 35 21)(4 36 22)(5 23 37)(6 24 38)(7 39 17)(8 40 18)(10 44 25)(12 27 46)(14 48 29)(16 31 42)
(1 15)(2 10)(3 13)(4 16)(5 11)(6 14)(7 9)(8 12)(17 43 39 32)(18 46 40 27)(19 41 33 30)(20 44 34 25)(21 47 35 28)(22 42 36 31)(23 45 37 26)(24 48 38 29)
(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)| (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41), (2,20,34)(4,36,22)(6,24,38)(8,40,18)(9,43,32)(10,25,44)(11,26,45)(12,46,27)(13,47,28)(14,29,48)(15,30,41)(16,42,31), (1,19,33)(2,20,34)(3,35,21)(4,36,22)(5,23,37)(6,24,38)(7,39,17)(8,40,18)(10,44,25)(12,27,46)(14,48,29)(16,31,42), (1,15)(2,10)(3,13)(4,16)(5,11)(6,14)(7,9)(8,12)(17,43,39,32)(18,46,40,27)(19,41,33,30)(20,44,34,25)(21,47,35,28)(22,42,36,31)(23,45,37,26)(24,48,38,29), (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( (1,7,5,3)(2,8,6,4)(9,15,13,11)(10,16,14,12)(17,37,21,33)(18,38,22,34)(19,39,23,35)(20,40,24,36)(25,42,29,46)(26,43,30,47)(27,44,31,48)(28,45,32,41), (2,20,34)(4,36,22)(6,24,38)(8,40,18)(9,43,32)(10,25,44)(11,26,45)(12,46,27)(13,47,28)(14,29,48)(15,30,41)(16,42,31), (1,19,33)(2,20,34)(3,35,21)(4,36,22)(5,23,37)(6,24,38)(7,39,17)(8,40,18)(10,44,25)(12,27,46)(14,48,29)(16,31,42), (1,15)(2,10)(3,13)(4,16)(5,11)(6,14)(7,9)(8,12)(17,43,39,32)(18,46,40,27)(19,41,33,30)(20,44,34,25)(21,47,35,28)(22,42,36,31)(23,45,37,26)(24,48,38,29), (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([[(1,7,5,3),(2,8,6,4),(9,15,13,11),(10,16,14,12),(17,37,21,33),(18,38,22,34),(19,39,23,35),(20,40,24,36),(25,42,29,46),(26,43,30,47),(27,44,31,48),(28,45,32,41)], [(2,20,34),(4,36,22),(6,24,38),(8,40,18),(9,43,32),(10,25,44),(11,26,45),(12,46,27),(13,47,28),(14,29,48),(15,30,41),(16,42,31)], [(1,19,33),(2,20,34),(3,35,21),(4,36,22),(5,23,37),(6,24,38),(7,39,17),(8,40,18),(10,44,25),(12,27,46),(14,48,29),(16,31,42)], [(1,15),(2,10),(3,13),(4,16),(5,11),(6,14),(7,9),(8,12),(17,43,39,32),(18,46,40,27),(19,41,33,30),(20,44,34,25),(21,47,35,28),(22,42,36,31),(23,45,37,26),(24,48,38,29)], [(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.PSU3(F2) ►in GL10(F73)
72 | 2 | 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 | 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 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 1 | 72 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 72 |
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 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 1 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 72 | 0 |
71 | 35 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
52 | 2 | 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 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 |
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 | 1 | 0 | 0 |
61 | 12 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
67 | 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 | 1 | 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 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |
G:=sub<GL(10,GF(73))| [72,72,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72],[1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,72,72,0,0,0,0,0,0,0,0,1,0],[71,52,0,0,0,0,0,0,0,0,35,2,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0],[61,67,0,0,0,0,0,0,0,0,12,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0] >;
C4.PSU3(F2) in GAP, Magma, Sage, TeX
C_4.{\rm PSU}_3({\mathbb F}_2)
% in TeX
G:=Group("C4.PSU(3,2)");
// GroupNames label
G:=SmallGroup(288,393);
// by ID
G=gap.SmallGroup(288,393);
# by ID
G:=PCGroup([7,-2,-2,-2,-2,-2,-3,3,28,309,92,219,100,346,80,9413,2028,691,12550,1581,2372]);
// Polycyclic
G:=Group<a,b,c,d,e|a^4=b^3=c^3=d^4=1,e^2=a^-1*d^2,a*b=b*a,a*c=c*a,d*a*d^-1=a^-1,a*e=e*a,e*c*e^-1=b*c=c*b,d*b*d^-1=c^-1,e*b*e^-1=b^-1*c,d*c*d^-1=b,e*d*e^-1=a*d^-1>;
// generators/relations
Export
Subgroup lattice of C4.PSU3(F2) in TeX
Character table of C4.PSU3(F2) in TeX