Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C23×C4 |
| direct product | C3×#2381 |
| 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 | |
424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 31 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 31 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,20)(17,19)(18,23)(22,28)(24,26)(25,27)(29,30,31);(7,12)(9,11)(13,15,17,23)(14,18,19,21)(16,24,25,22)(20,28,27,26);(5,7,8,12)(6,9,10,11)(14,19)(18,21)(20,27)(26,28);(6,10)(9,11)(13,16,17,25)(14,20,19,27)(15,22,23,24)(18,26,21,28)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16821);