Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
| direct product | Q8×ΓL(1,𝔽25) |
| Fitting | C3×C8×Q8 ⋊ C2 |
| Sylow | C3 ⋊ Q8×M16 |
388 conjugacy classes of subgroups; only the 195 normal subgroups are shown.
This group has 388 conjugacy classes of subgroups — too many to draw. Only its 195 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 192 |
A minimal faithful permutation representation — 4 generators on 19 points:
(1,2,3)(12,13,14,17,15,16,18,19);(2,3)(13,16)(17,19);(4,6,7,11)(5,8,9,10);(4,5,7,9)(6,10,11,8)90 conjugacy classes — view the full character table.
| maximal subgroup of | #10766105, #10766111, #10766116, … |
| maximal normal subgroup of | #10766105, #10766111, #10766116, … |
| maximal quotient of | C4⋊C4×ΓL(1,𝔽25), #10793834, Q8×#815, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 12495);