Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#2061 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
230 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 92 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) = 96 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,5,7)(3,8,4,6)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,24,22,23)(25,26,27);(1,3,5,4)(2,6,7,8)(9,11,12,19)(10,14,16,18)(13,22,21,20)(15,24,17,23);(1,4,5,3)(2,6,7,8)(10,15)(13,21)(14,23)(16,17)(18,24)(20,22)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5352);