Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C35 · Schur multiplier (H2) = C310 |
| direct product | C32×#6893 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| prime radical | C33×H(3,𝔽3) | C3 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
7892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2,3);(4,5,6);(7,8,10)(9,12,14)(11,15,13)(16,17,20)(18,22,28)(19,23,27)(21,29,26)(24,31,36)(25,32,35)(30,37,34)(33,39,41)(38,42,40);(17,21,23)(20,27,29)(22,30,32)(28,35,37)(31,38,39)(36,41,42);(8,11,12)(10,14,15)(16,18,24,19,25,33,26,34,40)(17,22,31,23,32,39,21,30,38)(20,28,36,27,35,41,29,37,42)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9114);