Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1722 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #127 |
| commutator, Frattini, cosocle | C42 | C23×C3 |
| prime radical | C3×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ G | |
198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,5)(3,6)(7,8)(9,10,13,16)(11,17,18,24)(12,20,21,15)(14,22,23,19)(25,26,27);(9,11,12,19,13,18,21,22)(10,14,15,17,16,23,20,24);(1,2,6,8,4,7,3,5)(11,18)(14,23)(17,24)(19,22)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5013);