Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×#135 |
| unambiguous semidirect product | (C22×#135)⋊C3 |
| Fitting, Sylow | C22×#135 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×A4 |
| commutator, cosocle | C22×C4 | C23×C3 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#135 | C3 |
| socle | C23 | C2×C3×D4 |
| hypercenter | #135 | A4 |
| hypocenter | C22 | C3×#135 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ #135 | |
354 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 354 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)(5,18,9,8)(6,12,13,11)(7,16,17,15)(10,20,19,14);(1,4)(2,3)(5,7)(6,10)(8,15)(9,17)(11,14)(12,20)(13,19)(16,18);(5,6)(7,14)(8,11)(9,13)(10,16)(12,18)(15,19)(17,20)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#359, A4×#360, A4×#443, … |
| maximal normal subgroup of | A4×#359, A4×#360, A4×#443, … |
| maximal quotient of | A4×#416, A4×#427, C4×A4×D8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18209);