Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #560 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#2382 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C24 |
| commutator | C22 | C23×C3×C4 |
| Frattini, cosocle | C23 | C24×C3 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
408 conjugacy classes of subgroups; only the 264 normal subgroups are shown.
This group has 408 conjugacy classes of subgroups — too many to draw. Only its 264 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 4 generators on 23 points:
(1,2,3,4)(6,9)(7,11)(14,15)(18,19)(21,22,23);(5,6)(7,12)(8,9)(10,11)(13,14,17,15)(16,19,20,18);(6,9)(10,12)(13,15,17,14)(16,19,20,18);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,18,15,19)114 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16822);