Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1705 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#1709 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#493 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,3)(4,5)(7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,23)(22,24)(25,26,27);(1,2,5,7,6,8,4,3)(9,11,12,18,13,19,21,24)(10,14,16,17,15,22,20,23);(1,3,4,8,6,7,5,2)(10,15)(14,22)(16,20)(17,23)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5000);