Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1468 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C3×#1472 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | C2×C22⋊C4 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C24×C3 | C2×C4 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C23×C3×C4 ◁ G | |
270 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 270 conjugacy classes of subgroups — too many to draw. Only its 96 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 — 3 generators on 27 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,15,13,16,20,24)(11,14,18,22,19,23,17,21)(25,26,27);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,23)(17,20)(21,24);(9,12,13,20)(10,15,16,24)(11,17,19,18)(14,21,23,22)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4763);