Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7320 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #342 |
| commutator | C24 | C42×C19 |
| Frattini, cosocle | C2×C4×D4 | C22×C19 |
| prime radical | C22×D4×C19 | C2×C4 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×D4×C19 ◁ G | |
342 conjugacy classes of subgroups; only the 78 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 35 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 2 generators on 35 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,25,23,24,26,27)(29,32)(33,34);(21,24)(22,26)(28,29)(30,34,35,32)(31,33)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 308);