Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×C3×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5);(7,10)(8,12)(14,15,17,19)(16,20,21,18);(7,10)(11,13)(14,16,17,21)(15,18,19,20);(6,7)(8,11)(9,10)(12,13)(16,21)(18,20);(1,2)(6,8)(7,11)(9,12)(10,13)(14,15)(16,20)(17,19)(18,21)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19986);