Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | D3×H(3,𝔽4) |
| Fitting | C3×H(3,𝔽4) ⋊ C2 |
| Sylow | C3 ⋊ C2×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C24×D3 |
| commutator, socle, cosocle | C22×C3 | C25 |
| Fitting | C3×H(3,𝔽4) | C2 |
| hypercenter | H(3,𝔽4) | D3 |
| hypocenter | C3 | C2×H(3,𝔽4) |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ H(3,𝔽4) | |
1408 conjugacy classes of subgroups; only the 459 normal subgroups are shown.
This group has 1408 conjugacy classes of subgroups — too many to draw. Only its 459 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 — 5 generators on 19 points:
(1,2,3)(5,9)(6,11)(13,17)(16,19);(6,11)(8,10)(13,17)(14,18);(4,6,7,11)(5,8,9,10)(12,14)(13,16)(15,18)(17,19);(4,5,7,9)(6,10,11,8)(12,13)(14,16)(15,17)(18,19);(2,3)57 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1655, D3×#1659, #10776400, … |
| maximal normal subgroup of | D3×#1655, D3×#1659, #10776400, … |
| maximal quotient of | D3×#2375, D3×#2379, #10957889, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19665);