Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4×C19 · Schur multiplier (H2) = C27×C4 |
| direct product | C19×#34186 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C25 |
| commutator | C22 | C24×C4×C19 |
| Frattini, cosocle | C23 | C25×C19 |
| prime radical | C23×D4×C19 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ G | |
2858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 5 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,23)(24,25,28,26)(27,30,31,29)(34,38)(36,39);(25,26)(29,30)(32,33,35,37)(34,36,38,39);(32,34,35,38)(33,36,37,39);(24,26,28,25)(27,30,31,29);(24,27)(25,29)(26,30)(28,31)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 27174);