Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1298 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C23×C4 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C24×C3 | Q8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C23×C3×C4 ◁ G | |
190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,17,14,18,13,16)(11,19,20,15,22,23,21,24)(25,26,27);(1,3)(2,5,6,7)(4,8)(9,11,13,21,14,22,12,20)(10,15,17,23,18,24,16,19)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 434);