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) = C24 |
| direct product | C3×#1907 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #121 |
| commutator | C2×C4 | C22×C3×C4 |
| Frattini, cosocle | C22×C4 | C23×C3 |
| prime radical | C2×C3×AGL(1,ℤ/8) | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ G | |
332 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 332 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)(5,6,9,8,10,12,18,16)(7,11,15,19,14,20,17,13)(21,22,23);(1,3,4,2)(5,7,9,17,10,14,18,15)(6,11,16,19,12,20,8,13);(5,8)(6,9)(7,13)(10,16)(11,17)(12,18)(14,19)(15,20)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5198);