Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | D4×C42⋊C3 |
| unambiguous semidirect product | C4⋊(C2×C42⋊C3) |
| unambiguous semidirect product | (C42×D4)⋊C3 |
| Fitting, Sylow | C42×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C22×C42⋊C3 |
| commutator, cosocle | C2×C42 | C22×C3 |
| Frattini, socle | C23 | C22×A4 |
| Fitting | C42×D4 | C3 |
| hypercenter | D4 | C42⋊C3 |
| hypocenter | C42 | C3×D4 |
| LCS | G ▷ C2×C42 ▷ C42 | |
| UCS | C1 ◁ C2 ◁ D4 | |
194 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 24 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:
(2,4)(5,6,9)(7,10,14)(8,11,13)(12,15,16);(1,2)(3,4)(5,12,8,7)(9,14,13,16)40 conjugacy classes — view the full character table.
| maximal subgroup of | D8×C42⋊C3, SD16×C42⋊C3, #33078⋊C3, … |
| maximal normal subgroup of | D8×C42⋊C3, SD16×C42⋊C3, #33078⋊C3, … |
| maximal quotient of | C22⋊C4×C42⋊C3, C4⋊C4×C42⋊C3, D4×#95⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5738);