Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C4×2−1+4⋊C3 |
| unambiguous semidirect product | (C4×2−1+4)⋊C3 |
| Fitting, Sylow | C4×2−1+4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×A4 |
| commutator, hypocenter | Q8 | C22×C3×C4 |
| Frattini, socle | C22 | C23×A4 |
| Fitting | C4×2−1+4 | C3 |
| prime radical | C2×2−1+4⋊C3 | C2 |
| cosocle | C2×Q8 | C23×C3 |
| hypercenter | C4×D4 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C4 ◁ C4×D4 | |
372 conjugacy classes of subgroups; only the 86 normal subgroups are shown.
This group has 372 conjugacy classes of subgroups — too many to draw. Only its 86 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(6,10)(7,8,13)(11,18,19,17,12,20)(14,16,15);(5,19,9,20)(6,13,10,15)(7,18,14,12)(8,11,16,17);(1,2,3,4)76 conjugacy classes — view the full character table.
| maximal subgroup of | #17439⋊C3, C4×#577⋊C3, C4×#578⋊C3, … |
| maximal normal subgroup of | #17439⋊C3, C4×#577⋊C3, C4×#578⋊C3, … |
| maximal quotient of | C4×#520⋊C3, C4×#519⋊C3, C4×#548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18157);