Finite Group DatabasebetaFinite Group Databasebeta| order | 12393 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C32×C9×C17 · Schur multiplier (H2) = C32×C9 |
| direct product | C17×#10624550 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C33×C17 | C33 |
| commutator | C32 | C32×C9×C17 |
| Frattini, cosocle | C33 | C33×C17 |
| LCS | G ▷ C32 ▷ C1 | |
| UCS | C1 ◁ C33×C17 ◁ G | |
374 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)(11,15,17)(12,16,18)(20,21,25)(23,26,24)(28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44);(10,11,16,14,17,18,13,15,12)(19,20,23,22,21,24,27,25,26);(10,12,15,13,18,17,14,16,11)(19,21,26,22,25,23,27,20,24)2601 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(12393, 110);