Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #396⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×#396⋊C3 |
| unambiguous semidirect product | (C2×#396)⋊C3 |
| Fitting, Sylow | C2×#396 ⋊ C3 |
| derived | #95 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #126⋊C3 |
| commutator, cosocle, hypocenter | #95 | C22×C3 |
| Frattini | C23 | C22×A4 |
| Fitting | C2×#396 | 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) = 14 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,2)(3,4,8)(5,11,14,6,10,12)(7,9,13);(1,2)(4,11)(5,6)(8,14,13,12)(9,10)36 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#396⋊C3, #14331⋊C3, #14562⋊C3, … |
| maximal normal subgroup of | C4×#396⋊C3, #14331⋊C3, #14562⋊C3, … |
| maximal quotient of | C2×#1443⋊C3, C4×#396⋊C3, #14748⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5761);