Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C17×C27 · Schur multiplier (H2) = C32 |
| direct product | C17×#10624444 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C9×C17 | H(3,𝔽3) |
| commutator | C32 | C3×C17×C27 |
| Frattini, cosocle | C32×C9 | C32×C17 |
| socle | C32×C17 | C32⋊C9 |
| prime radical | C34×C17 | C9 |
| LCS | G ▷ C32 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C3×C9×C17 ◁ C32×C9×C17 ◁ G | |
316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 53 |
| transitive degree | μt(G) = 1377 |
A minimal faithful permutation representation — 2 generators on 53 points:
(1,2,4)(3,5,8)(6,9,7)(10,11,14,12,15,20,17,22,27,13,16,21,18,23,28,25,30,33,19,24,29,26,31,34,32,35,36)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(2,5,6)2601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 4);