Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = S4×B(4,𝔽2) (order 1536) · Inn(G) = C22×A4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | C2×D4×A4 |
| unambiguous semidirect product | C4⋊(C22×A4) |
| unambiguous semidirect product | (C23×D4)⋊C3 |
| Fitting, Sylow | C23×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×A4 |
| commutator, cosocle | C23 | C23×C3 |
| Frattini | C2 | C23×A4 |
| Fitting | C23×D4 | C3 |
| socle | C24 | C22×C3 |
| hypercenter | C2×D4 | A4 |
| hypocenter | C22 | C2×C3×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
317 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 57 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 10 points:
(1,2)(4,5,6)(7,8)(9,10);(1,2)(3,6)(4,5)(8,10);(1,2)40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1497);