Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2 |
| direct product | C3×#1313 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C42 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C4×C8 | C22×C3 |
| socle, prime radical | C22×C3 | #97 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
80 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 2 generators on 27 points:
(9,10,11,15,12,17,19,23,13,16,18,24,20,14,21,22)(25,26,27);(1,2,3,5,4,6,7,8)(10,14,16,17)(11,18)(15,23,24,22)(19,21)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 449);