Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | A4×2−1+4 |
| Fitting, Sylow | C22×2−1+4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×A4 |
| commutator, socle, cosocle | C23 | C24×C3 |
| Fitting | C22×2−1+4 | C3 |
| hypercenter | 2−1+4 | A4 |
| hypocenter | C22 | C3×2−1+4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1222 conjugacy classes of subgroups; only the 204 normal subgroups are shown.
This group has 1222 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 20 points:
(2,3,4)(6,12)(10,13)(11,20)(18,19);(1,4)(2,3)(5,17,9,8)(6,20,12,11)(7,16,15,14)(10,19,13,18);(5,15,9,7)(6,13,12,10)(8,14,17,16)(11,18,20,19);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)68 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#454, A4×#456, A4×#577, … |
| maximal normal subgroup of | A4×#454, A4×#456, A4×#577, … |
| maximal quotient of | A4×#519, A4×#520, A4×#539, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20109);