Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C4 |
| direct product | C3×#1307 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #95 |
| commutator | C23 | C3×C42 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,4,7,5,6,8,3)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22)(25,26,27);(1,3,4,6,5,7,8,2)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 443);