Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#1603 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C3×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
244 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 244 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 — 3 generators on 27 points:
(5,8)(9,10,13,17)(11,18,21,24)(12,15,22,16)(14,19,23,20)(25,26,27);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,19)(13,21)(15,18)(16,24)(17,23)(20,22);(1,3)(2,5)(4,7)(6,8)(9,12,13,22)(10,15,17,16)(11,19,21,20)(14,18,23,24)69 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4894);