Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | A4×#126 |
| Fitting, Sylow | C22×#126 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×A4 |
| commutator, socle, cosocle | C24 | C23×C3 |
| Fitting | C22×#126 | C3 |
| prime radical | C23×A4 | C22 |
| hypercenter | #126 | A4 |
| hypocenter | C22 | C3×#126 |
| LCS | G ▷ C24 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ #126 | |
320 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 320 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(6,10)(9,12)(13,20,16,15)(14,19,17,18);(1,4)(2,3)(5,11,8,7)(6,12,10,9)(14,17)(18,19);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×#126, A4×#529, A4×#538, … |
| maximal normal subgroup of | C2×A4×#126, A4×#529, A4×#538, … |
| maximal quotient of | A4×#383, A4×#387, A4×#388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18166);