Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #392⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#399⋊C3 |
| unambiguous semidirect product | (C2×#399)⋊C3 |
| Fitting, Sylow | C2×#399 ⋊ C3 |
| derived | #95 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #127⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#399 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ #95 | |
| UCS | C1 ◁ C22 | |
213 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 213 conjugacy classes of subgroups — too many to draw. Only its 31 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 — 2 generators on 20 points:
(1,2,4)(5,7,6)(9,11,15)(10,13,17)(12,18,19,14,16,20);(1,7,5,2)(3,6,8,4)(9,10)(11,18,13,16)(12,14)(15,19,17,20)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#399⋊C3, #14322⋊C3, #14618⋊C3, … |
| maximal normal subgroup of | C4×#399⋊C3, #14322⋊C3, #14618⋊C3, … |
| maximal quotient of | C2×#1444⋊C3, C4×#399⋊C3, #14836⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5762);