Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1462 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#1820 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C22⋊C4 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C24 | C23×C3 |
| prime radical | C24×C3 | C23 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
300 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 104 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,27);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5111);