Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C8×C16×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#7136 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C8×C19 | C22 |
| commutator | C2 | C8×C16×C19 |
| Frattini, cosocle | C2×C4×C8 | C22×C19 |
| socle, prime radical | C23×C19 | C4×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8×C19 ◁ G | |
306 conjugacy classes of subgroups; only the 222 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 2 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(32,33,34,37,35,38,40,43,36,39,41,44,42,45,46,47);(21,23)(24,25,26,28,27,29,30,31)3040 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 124);