Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#2010 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | #127 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C2×C3 | #392 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4 ◁ C22×C3×C4 ◁ G | |
394 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 394 conjugacy classes of subgroups — too many to draw. Only its 88 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 — 3 generators on 19 points:
(1,2)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,19);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14);(6,15)(9,16)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5301);