Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C17 · Schur multiplier (H2) = C3 |
| direct product | C17×#10624483 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C32×C17 | #732 |
| commutator, Frattini, cosocle | C32×C9 | C32×C17 |
| prime radical | C33×C17 | H(3,𝔽3) |
| LCS | G ▷ C32×C9 ▷ C33 ▷ C3 ▷ C1 | |
| UCS | C1 ◁ C32×C17 ◁ C33×C17 ◁ C32×C9×C17 ◁ G | |
132 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)(10,11,14,20,31,23,13,17,24)(12,16,28,21,33,34,19,30,22)(15,25,35,26,32,36,27,18,29)(37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53);(2,5,6)(10,11,15,13,17,27,20,31,26)(12,16,23,19,30,14,21,33,24)(18,29,34,32,36,28,25,35,22)969 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 43);