Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×C3×H(3,ℤ/4) |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C42 |
| commutator | C4 | C2×C3×C42 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| socle | C22×C3 | #95 |
| prime radical | C24×C3 | D4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ G | |
404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2)(3,4,5);(7,11,12,17)(9,16)(13,20,19,18)(14,21);(6,7,9,13)(8,11,14,18)(10,12,16,19)(15,17,21,20)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5021);