Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C22 |
| direct product | C2×C3×SD64 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | D16 |
| commutator, Frattini, cosocle | C16 | C23×C3 |
| prime radical | C2×C3×D16 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C2×C3×C4 ◁ C2×C3×C8 ◁ C2×C3×C16 ◁ G | |
112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(35,36,37);(3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34);(4,18)(5,33)(6,16)(7,31)(8,14)(9,29)(10,12)(11,27)(13,25)(15,23)(17,21)(20,34)(22,32)(24,30)(26,28)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5551);