Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
| direct product | C22×A4⋊C8 |
| derived | A4 ⋊ C22×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | S4 |
| commutator, hypocenter | A4 | C22×C8 |
| Frattini | C4 | C22×S4 |
| Fitting | C24×C4 | D3 |
| socle | C25 | Dic3 |
| prime radical | C23×A4 | C4 |
| cosocle | C4×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C22×C4 | |
403 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 16 points:
(5,6,8,7)(9,10,11,13,12,14,15,16);(3,4)(6,7,8);(1,2)80 conjugacy classes — view the full character table.
| maximal subgroup of | #11711285, #11711286, #11711297, … |
| maximal normal subgroup of | #11711285, #11711286, #11711297, … |
| maximal quotient of | C22×SL(2,𝔽3)⋊C8, C2×#73068, C2×#73056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18052);