Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | A4×AGL(1,ℤ/8) |
| unambiguous semidirect product | (C22×AGL(1,ℤ/8))⋊C3 |
| Fitting, Sylow | C22×AGL(1,ℤ/8) ⋊ C3 |
400 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 400 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 12 points:
(2,3,4)(6,10)(9,11);(1,4)(2,3)(6,9)(7,12)(10,11);(5,6)(7,9)(8,10)(11,12)44 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#453, A4×#454, A4×#471, … |
| maximal normal subgroup of | A4×#453, A4×#454, A4×#471, … |
| maximal quotient of | A4×#418, A4×#417, A4×#428, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18210);