Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C42 · Schur multiplier (H2) = C22 |
| direct product | C3×#1301 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C22⋊C4 |
| commutator | C2×C4 | C3×C42 |
| Frattini, cosocle | C2×C42 | C22×C3 |
| socle | C22×C3 | #95 |
| prime radical | C23×C3 | C4⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
148 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 84 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:
(2,6)(3,4)(7,8)(9,10,12,15,13,16,20,23)(11,17,19,24,18,14,21,22)(25,26,27);(1,2,4,7,5,8,3,6)(9,11)(10,14,16,17)(12,19)(13,18)(15,22,23,24)(20,21)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 437);