Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1685 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×#1700 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #121 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
146 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 146 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) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,4,5)(3,7,8,6)(9,10)(11,17)(12,14)(13,21)(15,19)(16,20)(18,24)(22,23)(25,26,27);(1,2,3,6,4,5,8,7)(10,14)(15,23)(16,21)(17,24);(2,5)(6,7)(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4991);