Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×#5415 |
| Fitting | C23×D4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | GL(2,ℤ/4) |
| commutator, cosocle | C4×A4 | C23 |
| Frattini | C4 | C22×S4 |
| Fitting | C23×D4 | D3 |
| socle | C24 | #67 |
| hypercenter | C2×D4 | S4 |
| hypocenter | A4 | C2×D8 |
| LCS | G ▷ C4×A4 ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 ◁ C2×D4 | |
391 conjugacy classes of subgroups; only the 44 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 44 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 14 points:
(3,4,6)(7,8)(9,14)(10,12)(11,13);(4,5)(8,11)(9,13)(12,14);(1,2)38 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5415, #11712805, #11713060, … |
| maximal normal subgroup of | C4×#5415, #11712805, #11713060, … |
| maximal quotient of | C2×#73088, C2×#73085, C2×#73092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18064);