Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×#362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3 | #102 |
| commutator | C8 | C22×C3×C4 |
| Frattini, cosocle | C2×C8 | C23×C3 |
| prime radical | C23×C3 | SD16 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C22×C3×C8 ◁ G | |
216 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 216 conjugacy classes of subgroups — too many to draw. Only its 104 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(3,4,5);(6,7,10,15,12,16,11,27)(8,17,19,34,21,35,20,31)(9,22,24,32,26,37,25,14)(13,28,29,18,30,23,33,36);(6,8,12,21)(7,13,16,30)(9,23,26,28)(10,20,11,19)(14,31,32,34)(15,33,27,29)(17,22,35,37)(18,25,36,24)96 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5438);