Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1653 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
194 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 194 conjugacy classes of subgroups — too many to draw. Only its 72 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:
(2,5)(3,7)(6,8)(9,10,14,18)(11,19,22,23)(12,15,13,17)(16,20,24,21)(25,26,27);(1,2,4,6)(3,8,7,5)(10,15)(12,13)(16,23)(17,18)(19,24)(20,21);(9,11,14,22)(10,16,18,24)(12,21,13,20)(15,23,17,19)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4944);