Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2190 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2190 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C3 | #447 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ C2×C3×C4 ◁ C2×C3×M16 ◁ G | |
226 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 226 conjugacy classes of subgroups — too many to draw. Only its 64 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:
(2,6)(3,9)(4,12)(7,14)(8,15)(10,13)(11,16)(17,18,19);(1,2)(3,10)(4,13)(5,7)(6,9)(8,12)(11,15)(14,16);(3,11)(6,14)(9,16)(10,15)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5481);