Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | A4×#131 |
| Fitting, Sylow | C22×#131 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×A4 |
| commutator, socle | C23 | C22×C3×C4 |
| Frattini | C4 | C23×A4 |
| Fitting | C22×#131 | C3 |
| prime radical | A4×#41 | C2 |
| cosocle | C22×C4 | C23×C3 |
| hypercenter | #131 | A4 |
| hypocenter | C22 | C3×#131 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C8 ◁ #131 | |
354 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 354 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) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,3,4)(7,13)(10,16)(14,19)(17,20);(1,4)(2,3)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20);(5,11,15,6,9,18,8,12)(7,17,19,10,13,20,14,16)80 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#350, A4×#443, A4×#444, … |
| maximal normal subgroup of | A4×#350, A4×#443, A4×#444, … |
| maximal quotient of | A4×#405, A4×#408, A4×#424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18202);