Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#5168 |
| Fitting | C22×C3×SD16 ⋊ C2 |
| Sylow | C3 ⋊ C2×#495 |
592 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 592 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 25 points:
(1,2)(3,4,5)(7,9)(10,12,14,18)(11,16,15,24)(13,22,23,20)(17,25,21,19);(1,2)(7,9)(10,11)(12,19)(13,21)(14,15)(16,20)(17,23)(18,25)(22,24);(4,5)(6,7,8,9)(11,15)(12,18)(17,21)(20,22);(1,2)72 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#5168, #10803628, #10803665, … |
| maximal normal subgroup of | C4×#5168, #10803628, #10803665, … |
| maximal quotient of | C2×#71255, C2×#71333, C2×#71256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 16427);