Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
| direct product | M16×S4 |
| Fitting | C22×M16 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C22×S4 |
| commutator | C2×A4 | C22×C4 |
| Fitting | C22×M16 | D3 |
| socle | C23 | C2×C4×D3 |
| prime radical | C22×S4 | C4 |
| cosocle | C4×A4 | C23 |
| hypercenter | M16 | S4 |
| hypocenter | A4 | C2×M16 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C4 ◁ M16 | |
297 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 48 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(1,2,4,3);(2,3,4)(6,9)(10,12);(5,6,7,10,8,9,11,12)50 conjugacy classes — view the full character table.
| maximal subgroup of | S4×#100, #11712896, S4×#101, … |
| maximal normal subgroup of | S4×#100, #11712896, S4×#101, … |
| maximal quotient of | M16×GL(2,𝔽3), #11711897, M16×#226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18017);