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) = C23 |
| direct product | C3×#1479 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C4 | C2×D4 |
| commutator | C4 | C22×C3×C8 |
| Frattini, cosocle | C42 | C23×C3 |
| socle | C22×C3 | C2×C22⋊C4 |
| prime radical | C23×C3 | C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ G | |
248 conjugacy classes of subgroups; only the 124 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 124 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) = 192 |
A minimal faithful permutation representation — 3 generators on 27 points:
(1,2,3,5,4,6,7,8)(9,10)(11,18)(12,15)(13,17)(14,19)(16,22)(20,23)(21,24)(25,26,27);(9,11,13,20)(10,14,17,24)(12,21,22,19)(15,18,16,23);(9,12,13,22)(10,15,17,16)(11,19,20,21)(14,23,24,18)132 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 4770);