Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q8×#239 |
| Fitting | C3×Q82 ⋊ C2 |
| Sylow | C3 ⋊ Q8×#41 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×Q82 | C2 |
| prime radical | C2×#239 | C22 |
| hypercenter | Q82 | D3 |
| hypocenter | C3 | Q8×#41 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ Q82 | |
1068 conjugacy classes of subgroups; only the 529 normal subgroups are shown.
This group has 1068 conjugacy classes of subgroups — too many to draw. Only its 529 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 19 points:
(1,2,3)(12,14,15,18)(13,16,17,19);(12,13,15,17)(14,19,18,16);(2,3)(14,18)(16,19);(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)75 conjugacy classes — view the full character table.
| maximal subgroup of | #10771162, #10771170, #10771177, … |
| maximal normal subgroup of | #10771162, #10771170, #10771177, … |
| maximal quotient of | C4⋊C4×#239, Q8×#892, #10954843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19615);