Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1281 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C22 | C3×C4×C8 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C23×C3 | C42 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22)(25,26,27);(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 417);