Finite Group DatabasebetaFinite Group Databasebeta| order | 384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| direct product | C2×A4×Q16 |
| unambiguous semidirect product | (C23×Q16)⋊C3 |
| Fitting, Sylow | C23×Q16 ⋊ C3 |
344 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 344 conjugacy classes of subgroups — too many to draw. Only its 66 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(4,5,6)(7,9,11,17)(8,12,14,21)(10,19,20,16)(13,22,18,15);(1,2)(3,6)(4,5)(7,8,11,14)(9,15,17,22)(10,18,20,13)(12,19,21,16);(1,2)56 conjugacy classes — view the full character table.
| maximal subgroup of | A4×#358, A4×#362, C4×A4×Q16, … |
| maximal normal subgroup of | A4×#358, A4×#362, C4×A4×Q16, … |
| maximal quotient of | C2×A4×#103, C2×A4×#107, C4×A4×Q16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(384, 18208);