Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C22×C3×#135 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C4 | D4 |
| commutator, Frattini, cosocle | C4 | C25×C3 |
| socle | C23×C3 | C2×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C4 ◁ C23×C3×C4 ◁ G | |
1504 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)(11,18)(14,20)(15,19)(16,22)(21,23);(8,9)(10,14)(11,19)(12,15)(13,18)(16,23)(17,21)(20,22);(1,2)(3,4)(8,10,12,17)(9,14,15,21)(11,16,18,22)(13,20,19,23)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 20037);