Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #534 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#2331 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,4,3)(5,6,8,10)(7,11,12,9)(14,16)(15,19)(21,22,23);(1,3,4,2)(13,14,17,16)(15,20,19,18);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,20)(17,19);(13,16,17,14)(15,20,19,18)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16771);