Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12715 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12718 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1781 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C19×#3498 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×H(5,𝔽2) ◁ G | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 304 |
A minimal faithful permutation representation — 3 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,31)(23,33,24,28)(25,29)(26,30)(27,32,35,34);(20,22,26,25)(21,27,30,35)(23,32,24,34)(28,29,33,31);(20,23,26,24)(21,27,30,35)(22,32,25,34)(28,31,33,29)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5706);