Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of S4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C42×S4 |
| Fitting | C22×C42 ⋊ D3 |
| derived | A4 ⋊ C2×C42 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C42 | S4 |
| commutator, hypocenter | A4 | C2×C42 |
| Frattini | C22 | C22×S4 |
| Fitting | C22×C42 | D3 |
| socle | C24 | C22×D3 |
| prime radical | C22×S4 | C22 |
| cosocle | C22×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C42 | |
497 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 497 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 48 |
A minimal faithful permutation representation — 3 generators on 12 points:
(9,10,12,11);(5,6,7,8)(10,11,12);(1,2,3,4)80 conjugacy classes — view the full character table.
| maximal subgroup of | C4×C8×S4, C4×#5400, #11712571, … |
| maximal normal subgroup of | C4×C8×S4, C4×#5400, #11712571, … |
| maximal quotient of | C42×SL(2,ℤ/4), C4×(C2×S4)⋊C4, C4×SL(2,ℤ/4)⋊C4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17862);