Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of B(4,𝔽2) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C24 |
| direct product | C3×#1890 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #120 |
| commutator | C23 | C22×C3×C4 |
| Frattini, cosocle | C24 | C23×C3 |
| prime radical | C22×C3×D4 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C24×C3 ◁ G | |
386 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 386 conjugacy classes of subgroups — too many to draw. Only its 112 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:
(1,2,4,3)(5,6)(7,10)(8,15,16,11)(9,12)(13,17,19,20)(14,18)(21,22,23);(1,3,4,2)(5,7,9,14)(6,10,12,18)(8,13,16,19)(11,17,15,20);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5181);