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) = C38 |
| direct product | C32×#6897 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C34 | C33 |
| commutator, Frattini, cosocle | C32 | C35 |
| prime radical | C35 | C32 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C34 ◁ G | |
| LECS | G ▷ C32 ▷ C1 | |
3932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 60 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 5 generators on 60 points:
(1,2,3);(4,5,6);(7,8,12)(9,16,29)(10,14,22)(11,15,23)(13,24,20)(17,26,32)(18,25,33)(19,27,30)(21,28,31)(34,35,38)(36,39,45)(37,41,46)(40,50,44)(42,47,53)(43,49,54)(48,57,52)(51,56,59)(55,60,58);(7,9,17,11,10,18,21,20,19)(8,13,25,15,16,27,28,14,26)(12,22,30,23,24,32,31,29,33)(34,36,42,37,43,51,44,52,58)(35,39,47,41,49,56,40,48,55)(38,45,53,46,54,59,50,57,60);(7,10,19,11,20,17,21,9,18)(8,14,27,15,13,26,28,16,25)(12,22,30,23,24,32,31,29,33)(35,40,41)(38,46,50)(39,48,49)(45,54,57)(47,55,56)(53,59,60)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 9156);