Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#101 |
| unambiguous semidirect product | (C22×#101)⋊C3 |
| Fitting, Sylow | C22×#101 ⋊ C3 |
186 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 186 conjugacy classes of subgroups — too many to draw. Only its 36 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 20 points:
(2,3,4)(5,15,9,7)(6,13,11,10)(8,14,17,16)(12,18,19,20);(1,4)(2,3)(5,12,17,6,9,19,8,11)(7,20,16,13,15,18,14,10)44 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#355, A4×#356, C2×A4×#101, … |
| maximal normal subgroup of | A4×#355, A4×#356, C2×A4×#101, … |
| maximal quotient of | A4×#324, #17430⋊C3, A4×#330, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5808);