Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(5,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C26×C3 · Schur multiplier (H2) = C214 |
| direct product | C2×C3×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C24 |
| commutator, Frattini, cosocle | C2 | C26×C3 |
| socle | C22×C3 | C25 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
5956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 6 generators on 21 points:
(1,2)(3,4,5);(7,11)(8,14)(13,18)(15,20);(6,7)(8,12)(9,13)(10,11)(14,17)(15,19)(16,18)(20,21);(7,11)(12,17)(13,18)(19,21);(6,8)(7,12)(9,15)(10,14)(11,17)(13,19)(16,20)(18,21);(1,2)(6,9,10,16)(7,13,11,18)(8,15,14,20)(12,19,17,21)204 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20159);