Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C28 |
| direct product | C2×C3×#564 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C24 |
| commutator, Frattini, cosocle | C22 | C25×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 5 generators on 37 points:
(1,2)(3,4,5);(8,14)(9,17)(11,15)(12,19)(22,23,26,27)(24,28,30,34)(25,32,33,29)(31,37,36,35);(6,7,10,13)(8,15,14,11)(9,12,17,19)(16,21,20,18)(23,27)(24,30)(25,33)(35,37);(6,8,10,14)(7,11,13,15)(9,16,17,20)(12,18,19,21)(22,24)(23,28)(25,31)(26,30)(27,34)(29,35)(32,37)(33,36);(1,2)(6,9)(7,12)(8,16)(10,17)(11,18)(13,19)(14,20)(15,21)(22,25,26,33)(23,29,27,32)(24,31,30,36)(28,35,34,37)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 19990);