Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C16 · Schur multiplier (H2) = C22 |
| direct product | C3×Q8×C16 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C16 | C22 |
| commutator | C2 | C22×C3×C16 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| socle, prime radical | C22×C3 | C22×C8 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C16 ◁ G | |
134 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3)(4,5,6,7)(8,11,10,9);(4,8,6,10)(5,9,7,11);(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5473);