Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1365 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C3 | #102 |
| commutator | C2×C8 | C2×C3×C4 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×C8 ◁ G | |
86 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(2,5)(3,7)(6,8)(9,10,12,15,14,18,11,19,13,17,20,23,21,24,22,16)(25,26,27);(1,2,4,6)(3,8,7,5)(10,15)(11,20)(12,22)(14,21)(16,18)(17,23)(19,24)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 501);