Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C8 · Schur multiplier (H2) = C2×C8 |
| direct product | C3×#1450 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C8 | C22 |
| commutator | C2 | C2×C3×C4×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C42 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C8 ◁ G | |
212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26,27);(1,2,3,5,4,6,7,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24)240 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4741);