Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| isoclinism | isoclinic class of #127⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C4×#127⋊C3 |
| unambiguous semidirect product | (C4×#127)⋊C3 |
| Fitting, Sylow | C4×#127 ⋊ C3 |
| derived | C42 ⋊ C2×C3×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | #127⋊C3 |
| commutator, hypocenter | C42 | C2×C3×C4 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | C4×#127 | C3 |
| prime radical | C2×#127⋊C3 | C2 |
| cosocle | C2×C42 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C4 | |
194 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 194 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) = 16 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 16 points:
(5,6,9)(7,15,14,12,10,16)(8,11,13);(1,2,3,4)(5,12,8,7)(9,14,13,16)40 conjugacy classes — view the full character table.
| maximal subgroup of | C8×#127⋊C3, #21681⋊C3, #23355⋊C3, … |
| maximal normal subgroup of | C8×#127⋊C3, #21681⋊C3, #23355⋊C3, … |
| maximal quotient of | C4×#384⋊C3, C4×#392⋊C3, C4×#399⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5749);