Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#5722 |
| Sylow | C3 ⋊ C2×Q82 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×C4⋊Q8 | C2 |
| socle, prime radical | C23×C3 | C24 |
| hypercenter | C2×C4⋊Q8 | D3 |
| hypocenter | C3 | C2×Q82 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×C4⋊Q8 | |
1112 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 581 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5)(6,8,9,13)(7,10,11,12)(16,17)(19,21);(6,7,9,11)(8,12,13,10)(14,17,18,16)(15,19,20,21);(14,16,18,17)(15,19,20,21);(4,5)(14,15,18,20)(16,21,17,19);(1,2)78 conjugacy classes — view the full character table.
| maximal subgroup of | #10686952, #10687303, #10687307, … |
| maximal normal subgroup of | #10686952, #10687303, #10687307, … |
| maximal quotient of | C2×#74323, C2×#75981, C2×#75987, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18779);