Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C2×#5717 |
| Fitting | C2×C3×#127 ⋊ C2 |
| Sylow | C3 ⋊ C2×D42 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C23×D3 |
| commutator, cosocle | C22×C3 | C25 |
| Frattini | C22 | C24×D3 |
| Fitting | C2×C3×#127 | C2 |
| socle | C23×C3 | C24 |
| hypercenter | C2×#127 | D3 |
| hypocenter | C3 | C2×D42 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C23 ◁ C2×#127 | |
2536 conjugacy classes of subgroups; only the 581 normal subgroups are shown.
This group has 2536 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) = 13 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 13 points:
(3,4,5)(6,7,8,9)(10,12,13,11);(1,2)(7,9)(10,11)(12,13);(7,9)(11,12);(4,5)(7,9);(1,2)78 conjugacy classes — view the full character table.
| maximal subgroup of | #10686887, #10687251, #10687255, … |
| maximal normal subgroup of | #10686887, #10687251, #10687255, … |
| maximal quotient of | C2×#76431, C2×#74360, C2×#76432, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18770);