Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #120⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×#120⋊C3 |
| unambiguous semidirect product | (C4×#120)⋊C3 |
| Fitting, Sylow | C4×#120 ⋊ C3 |
| derived | C24 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #120⋊C3 |
| commutator, hypocenter | C24 | C2×C3×C4 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | C4×#120 | C3 |
| prime radical | C2×#120⋊C3 | C2 |
| cosocle | C25 | C22×C3 |
| LCS | G ▷ C24 | |
| UCS | C1 ◁ C4 | |
254 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 254 conjugacy classes of subgroups — too many to draw. Only its 27 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 — 2 generators on 12 points:
(5,6)(7,10,11,9,8,12);(1,2,3,4)(6,12)(9,10)40 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#120⋊C3, #21440⋊C3, #23306⋊C3, … |
| maximal normal subgroup of | C8×#120⋊C3, #21440⋊C3, #23306⋊C3, … |
| maximal quotient of | C4×#379⋊C3, C4×#393⋊C3, C4×#395⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5747);