Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#2227 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C23×C3 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
130 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 130 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,6)(3,7,8,5)(10,14)(11,17)(13,16)(15,21)(18,24)(19,23)(20,22)(25,26,27);(1,3,4,8)(2,5,6,7)(9,10,12,14)(11,15,18,19)(13,16,20,22)(17,23,24,21);(9,11,12,18)(10,15,14,19)(13,21,20,23)(16,17,22,24)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5518);