Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C3×#2040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator, Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C3×#537 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
290 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 290 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) = 23 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 23 points:
(2,4)(5,6)(7,15)(8,13)(9,12)(10,14)(11,17)(16,20)(18,19)(21,22,23);(1,2)(3,4)(5,7)(6,11)(8,17)(9,18)(10,16)(12,20)(13,15)(14,19);(1,2,3,4)(6,12)(7,16)(9,10)(13,14)(17,18)(19,20)78 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5331);