Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | A4×#104 |
| unambiguous semidirect product | (C22×#104)⋊C3 |
| Fitting, Sylow | C22×#104 ⋊ C3 |
216 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 216 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) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(2,3,4)(5,7)(6,8)(9,11)(10,12);(1,4)(2,3)(5,7,8,6,10,12,9,11)56 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4×#104, A4×#421, A4×#443, … |
| maximal normal subgroup of | C2×A4×#104, A4×#421, A4×#443, … |
| maximal quotient of | A4×D4⋊C8, A4×Q8⋊C8, #17518⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 5815);