Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1347 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | C3×#1349 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #102 |
| commutator | C2×C8 | C2×C3×C4 |
| Frattini, cosocle | C22×C8 | C22×C3 |
| prime radical | C3×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
114 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 114 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,17,14,19,13,18)(11,20,22,16,23,15,21,24)(25,26,27);(1,3)(2,5)(4,8)(6,7)(10,15)(11,21)(12,13)(16,17)(18,24)(19,20)(22,23)87 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 485);