Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
| direct product | D3×#454 |
| Fitting | C3×#454 ⋊ C2 |
| Sylow | C3 ⋊ C2×#454 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#120 |
| commutator, cosocle | C2×C3×C4 | C24 |
| Frattini | C2×C4 | C23×D3 |
| Fitting | C3×#454 | C2 |
| socle | C2×C3 | C2×#120 |
| hypercenter | #454 | D3 |
| hypocenter | C3 | C2×#454 |
| LCS | G ▷ C2×C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #454 | |
728 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 728 conjugacy classes of subgroups — too many to draw. Only its 135 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 — 4 generators on 19 points:
(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,18,19);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15);(2,6)(4,11)(7,15)(8,12)(9,13)(14,16);(18,19)48 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1404, #10715440, D3×#2202, … |
| maximal normal subgroup of | D3×#1404, #10715440, D3×#2202, … |
| maximal quotient of | D3×#1597, D3×#1615, D3×#1598, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12859);