Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C3×#1579 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle, prime radical | C22×C3 | C4×D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C3×C4×C8 ◁ G | |
160 conjugacy classes of subgroups; only the 100 normal subgroups are shown.
This group has 160 conjugacy classes of subgroups — too many to draw. Only its 100 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 384 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,24)(21,23)(25,26,27);(9,11,13,19)(10,14,15,23)(12,21,22,18)(16,24,20,17);(1,2,3,5,4,6,7,8)(10,15)(14,23)(16,20)(17,24)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4870);