Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C25 |
| direct product | C2×C3×#340 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24×C3 | D4 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24×C3 ◁ C23×C3×C4 ◁ G | |
512 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(10,11,12,13)(15,18)(16,20)(19,21);(6,7,8,9)(14,15,16,18,17,19,20,21)168 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5025);