Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of S4 |
| automorphism | Aut(G) = S4×#120 (order 768) · Inn(G) = S4 · Out(G) = #120 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
| direct product | C2×C4×S4 |
| Fitting | C23×C4 ⋊ D3 |
| derived | A4 ⋊ C22×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | S4 |
| commutator, hypocenter | A4 | C22×C4 |
| Frattini | C2 | C22×S4 |
| Fitting | C23×C4 | D3 |
| socle | C24 | C2×D3 |
| prime radical | C22×S4 | C2 |
| cosocle | C2×A4 | C23 |
| LCS | G ▷ A4 | |
| UCS | C1 ◁ C2×C4 | |
233 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 43 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:
(3,4,6,5);(4,5,6)(7,8,9,10);(1,2)40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1469);