Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C16×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#7148 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C8×C19 | D4 |
| commutator | C4 | C4×C16×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle | C22×C19 | #336 |
| prime radical | C23×C19 | C22⋊C8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
302 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 134 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 2 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,24)(23,26)(25,27)(28,29,30,33,31,34,36,39,32,35,37,40,38,41,42,43);(21,24,26,25)2128 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 136);