Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×D3×#126 |
| Fitting | C2×C3×#126 ⋊ C2 |
| Sylow | C3 ⋊ C22×#126 |
1320 conjugacy classes of subgroups; only the 517 normal subgroups are shown.
This group has 1320 conjugacy classes of subgroups — too many to draw. Only its 517 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5)(6,8,9,13)(7,11,10,12)(15,19)(18,21);(7,10)(11,12)(14,16,17,20)(15,18,19,21);(6,7)(8,12)(9,10)(11,13)(14,15)(16,18)(17,19)(20,21);(4,5);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#1473, #10761280, C4×D3×#126, … |
| maximal normal subgroup of | D3×#1473, #10761280, C4×D3×#126, … |
| maximal quotient of | C2×D3×#383, C2×D3×#387, C2×D3×#388, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18687);