Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C22×C4⋊S4 |
| Fitting | C24×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C23 | C2×S4 |
| commutator, cosocle | C2×A4 | C24 |
| Frattini | C2 | C23×S4 |
| Fitting | C24×C4 | D3 |
| socle | C25 | C2×D3 |
| hypercenter | C22×C4 | S4 |
| hypocenter | A4 | C22×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C23 ◁ C22×C4 | |
1291 conjugacy classes of subgroups; only the 132 normal subgroups are shown.
This group has 1291 conjugacy classes of subgroups — too many to draw. Only its 132 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 — 4 generators on 12 points:
(5,6,8)(9,10,11,12);(6,7)(10,12);(3,4);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | #11711721, #11711731, #11711745, … |
| maximal normal subgroup of | #11711721, #11711731, #11711745, … |
| maximal quotient of | C22×C4⋊SL(2,ℤ/4), C22×(C2×S4)⋊C4, C2×C4×C4⋊S4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20049);