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×#2320 |
| 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 | |
444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,16)(21,22,23);(13,14,17,18)(15,20,16,19);(6,9)(10,12)(13,15,17,16)(14,19,18,20);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,19,18,20)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16760);