Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C92×C17 · Schur multiplier (H2) = C32×C9 |
| direct product | C9×C17×C9⋊C9 |
| unambiguous semidirect product | C9⋊(C92×C17) |
| unambiguous semidirect product | (C9×C17)⋊C92 |
| subgroup | G/H | |
|---|---|---|
| center | C32×C9×C17 | C32 |
| commutator | C3 | C3×C92×C17 |
| Frattini, cosocle | C33 | C33×C17 |
| socle, prime radical | C33×C17 | C33 |
| LCS | G ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C9×C17 ◁ G | |
482 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 12393 |
A minimal faithful permutation representation — 3 generators on 44 points:
(1,2,4,3,5,7,6,8,9)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(10,11,13,12,14,16,15,17,18)(20,22,25)(23,27,26);(19,20,23,21,22,26,24,25,27)5049 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 106);