Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×C3×#419 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C3 | C2×D4 |
| commutator | C4 | C23×C3×C4 |
| Frattini, cosocle | C2×C4 | C24×C3 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C23×C3×C4 ◁ G | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5);(6,7,8,9)(11,15)(13,21)(16,23)(17,22)(18,24)(19,25);(6,7)(8,9)(10,11,14,17)(12,18,20,16)(13,22,21,15)(19,23,25,24);(1,2)(10,12,14,20)(11,16,17,18)(13,19,21,25)(15,23,22,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 17334);