Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C25 |
| direct product | C22×#120⋊C3 |
| Fitting, Sylow | C22×#120 ⋊ C3 |
| derived | C24 ⋊ C23×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | #120⋊C3 |
| commutator, cosocle, hypocenter | C24 | C23×C3 |
| Frattini | C22 | C23×A4 |
| Fitting | C22×#120 | C3 |
| socle | C24 | C2×A4 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C22 | |
854 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 53 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(5,6)(7,10,11,9,8,12);(3,4)(6,12)(9,10);(1,2)40 conjugacy classes — view the full character table.
| maximal subgroup of | #14657⋊C3, #14694⋊C3, C2×C4×#120⋊C3, … |
| maximal normal subgroup of | #14657⋊C3, #14694⋊C3, C2×C4×#120⋊C3, … |
| maximal quotient of | C22×#379⋊C3, C2×C4×#120⋊C3, C22×#393⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18158);