Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1306 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #95 |
| commutator | C23 | C3×C42 |
| Frattini, cosocle | C23×C4 | C22×C3 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
192 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 192 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 96 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,18,14)(12,16)(13,15)(19,23,24,21)(20,22)(25,26,27);(1,3)(2,5,6,7)(4,8)(9,11,12,19,13,18,20,24)(10,14,16,21,15,17,22,23)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 442);