Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
| direct product | D3×#585 |
| Fitting | C3×#585 ⋊ C2 |
| Sylow | C3 ⋊ C2×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C24×D3 |
| commutator, socle, cosocle | C2×C3 | C26 |
| Frattini | C2 | C25×D3 |
| Fitting | C3×#585 | C2 |
| hypercenter | #585 | D3 |
| hypocenter | C3 | C2×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ #585 | |
5956 conjugacy classes of subgroups; only the 3203 normal subgroups are shown.
This group has 5956 conjugacy classes of subgroups — too many to draw. Only its 3203 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 6 generators on 19 points:
(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(17,18,19);(2,7)(6,9)(8,14)(13,15);(3,10)(6,9)(11,16)(13,15);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15);(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16);(18,19)102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1791, #10794464, D3×#1794, … |
| maximal normal subgroup of | D3×#1791, #10794464, D3×#1794, … |
| maximal quotient of | D3×#3428, C4×D3×H(5,𝔽2), C4×D3×2−1+4, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20149);