Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of C22⋊A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×C4×C22⋊A4 |
| Fitting, Sylow | C25×C4 ⋊ C3 |
| derived | C24 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | C22⋊A4 |
| commutator, hypocenter | C24 | C2×C3×C4 |
| Frattini | C2 | C22×C22⋊A4 |
| Fitting | C25×C4 | C3 |
| socle | C26 | C2×C3 |
| prime radical | C22×C22⋊A4 | C2 |
| cosocle | C25 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C2×C4 | |
1852 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 1852 conjugacy classes of subgroups — too many to draw. Only its 64 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 — 3 generators on 14 points:
(4,5,6)(8,9,10)(11,12,13,14);(1,2)(3,6)(4,5);(7,10)(8,9)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#379⋊C3, C4⋊(#379⋊C3), #14693⋊C3, … |
| maximal normal subgroup of | C4×#379⋊C3, C4⋊(#379⋊C3), #14693⋊C3, … |
| maximal quotient of | C42×C22⋊A4, C22⋊C4×C22⋊A4, C4⋊C4×C22⋊A4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20114);