Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C3×#2016 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ C2×C3×C42 ◁ G | |
370 conjugacy classes of subgroups; only the 116 normal subgroups are shown.
This group has 370 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2)(3,4)(9,10,14,18)(11,19,12,21)(13,23,24,17)(15,22,16,20)(25,26,27);(2,4)(5,6)(7,8)(9,11,14,12)(10,15,18,16)(13,22,24,20)(17,21,23,19);(6,8)(9,12,14,11)(10,16,18,15)(13,20,24,22)(17,19,23,21)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5307);