Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×#519⋊C3 |
| unambiguous semidirect product | (C2×#519)⋊C3 |
| Fitting, Sylow | C2×#519 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini | C22 | C23×A4 |
| Fitting | C2×#519 | C3 |
| prime radical | C23×SL(2,𝔽3) | C2 |
| cosocle | C2×Q8 | C23×C3 |
| hypercenter | C2×C22⋊C4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C23 ◁ C2×C22⋊C4 | |
403 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 22 points:
(3,4,5,6)(8,12)(9,10,15)(13,20,21,19,14,22)(16,18,17);(1,2)(7,21,11,22)(8,15,12,17)(9,20,16,14)(10,13,18,19);(1,2)76 conjugacy classes — view the full character table.
| maximal subgroup of | #17352⋊C3, #20355⋊C3, C4×#519⋊C3, … |
| maximal normal subgroup of | #17352⋊C3, #20355⋊C3, C4×#519⋊C3, … |
| maximal quotient of | C2×#73167, C4×#519⋊C3, C2×C22⋊C4×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18153);