Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#5688 |
| Fitting | C2×C3×#125 ⋊ C2 |
| Sylow | C3 ⋊ C2×#533 |
1168 conjugacy classes of subgroups; only the 541 normal subgroups are shown.
This group has 1168 conjugacy classes of subgroups — too many to draw. Only its 541 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) = 192 |
A minimal faithful permutation representation — 5 generators on 21 points:
(1,2)(3,4,5)(6,8,9,13)(7,10,11,12)(14,16)(15,18)(17,20)(19,21);(6,7,9,11)(8,12,13,10)(16,20)(18,21);(14,15,17,19)(16,21,20,18);(4,5)(16,20)(18,21);(1,2)84 conjugacy classes — view the full character table.
| maximal subgroup of | #10686666, #10761200, #10761319, … |
| maximal normal subgroup of | #10686666, #10761200, #10761319, … |
| maximal quotient of | C2×#74148, C2×#74938, C2×#75963, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18659);