Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#137 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C4 | C25×C3 |
| prime radical | C22×C3×#41 | C2 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 5 generators on 23 points:
(1,2)(5,6,7);(3,4);(9,13)(10,16)(11,18)(14,21)(15,19)(20,22);(8,9,12,15)(10,14,16,22)(11,19,18,13)(17,21,23,20);(1,2)(3,4)(8,10)(9,14)(11,17)(12,16)(13,20)(15,22)(18,23)(19,21)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20039);