Finite Group DatabasebetaFinite Group Databasebeta| order | 2187 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of C32⋊H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C33×C9 · Schur multiplier (H2) = C34 |
| direct product | C9×#6897 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9 | C33 |
| commutator | C32 | C33×C9 |
| Frattini, socle, cosocle | C33 | C34 |
| prime radical | C34 | C33 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C32×C9 ◁ G | |
| LECS | G ▷ C33 ▷ C1 | |
866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 63 |
| transitive degree | μt(G) = 729 |
A minimal faithful permutation representation — 4 generators on 63 points:
(1,2,3,4,5,6,7,8,9);(10,11,15)(12,19,32)(13,17,25)(14,18,26)(16,27,23)(20,29,35)(21,28,36)(22,30,33)(24,31,34)(37,38,41)(39,42,48)(40,44,49)(43,53,47)(45,50,56)(46,52,57)(51,60,55)(54,59,62)(58,63,61);(10,12,20,14,13,21,24,23,22)(11,16,28,18,19,30,31,17,29)(15,25,33,26,27,35,34,32,36)(37,39,45,40,46,54,47,55,61)(38,42,50,44,52,59,43,51,58)(41,48,56,49,57,62,53,60,63);(10,13,22,14,23,20,24,12,21)(11,17,30,18,16,29,31,19,28)(15,25,33,26,27,35,34,32,36)(38,43,44)(41,49,53)(42,51,52)(48,57,60)(50,58,59)(56,62,63)459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(2187, 5903);