Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1655 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C3×#1657 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#496 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
206 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 206 conjugacy classes of subgroups — too many to draw. Only its 60 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,5)(3,7)(6,8)(10,14)(12,19)(15,23)(16,20)(17,21)(22,24)(25,26,27);(3,7)(5,8)(9,10)(11,15)(12,20)(13,16)(14,19)(17,22)(18,24)(21,23);(1,2)(3,8)(4,6)(5,7)(9,11,13,18)(10,15,16,24)(12,21,19,17)(14,22,20,23)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4948);