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) = C42 |
| direct product | C3×#1274 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C42 |
| commutator | C4 | C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C23×C3 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,23)(12,15)(13,16)(14,18,21,24)(20,22)(25,26,27);(9,11)(10,14,16,21)(12,18)(13,19)(15,17,22,23)(20,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 410);