Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C2×C4 |
| direct product | C3×#1930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C23×C3 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
212 conjugacy classes of subgroups; only the 96 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 192 |
A minimal faithful permutation representation — 3 generators on 23 points:
(1,2,4,3)(6,10)(7,13)(8,15)(11,18)(12,17)(16,19)(21,22,23);(1,3,4,2)(5,6,8,10,9,12,15,17)(7,11,14,16,13,19,20,18);(5,7,8,14,9,13,15,20)(6,11,10,16,12,19,17,18)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5221);