Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| automorphism | Aut(G) = S4×B(4,𝔽2) (order 1536) · Inn(G) = C2×S4 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
| direct product | C2×C4⋊S4 |
| Fitting | C23×C4 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C2×S4 |
| commutator, cosocle | C2×A4 | C23 |
| Frattini | C2 | C22×S4 |
| Fitting | C23×C4 | D3 |
| socle | C24 | C2×D3 |
| hypercenter | C2×C4 | S4 |
| hypocenter | A4 | C2×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
233 conjugacy classes of subgroups.
| minimal degree | μ(G) = 10 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 10 points:
(3,4,6)(7,8,9,10);(4,5)(8,10);(1,2)28 conjugacy classes — view the full character table.
| maximal subgroup of | #73053, #73113, #73116, … |
| maximal normal subgroup of | #73053, #73113, #73116, … |
| maximal quotient of | #85283, C4×C4⋊S4, #85289, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1470);