Finite Group DatabasebetaFinite Group Databasebeta| order | 192 |
| isoclinism | isoclinic class of A4 |
| automorphism | Aut(G) = C2×D4×S4 (order 384) · Inn(G) = A4 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C2×C3×C8 · Schur multiplier (H2) = C22 |
| direct product | C2×C8×A4 |
| unambiguous semidirect product | (C23×C8)⋊C3 |
| Fitting, Sylow | C23×C8 ⋊ C3 |
| derived | C22 ⋊ C2×C3×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C8 | A4 |
| commutator, hypocenter | C22 | C2×C3×C8 |
| Frattini | C4 | C22×A4 |
| Fitting | C23×C8 | C3 |
| socle | C24 | C3×C4 |
| prime radical | C22×A4 | C4 |
| cosocle | C22×C4 | C22×C3 |
| LCS | G ▷ C22 | |
| UCS | C1 ◁ C2×C8 | |
93 conjugacy classes of subgroups.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 14 points:
(4,5,6)(7,8,9,11,10,12,13,14);(1,2)(3,6)(4,5)64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×A4⋊C16, #73104, #73105, … |
| maximal normal subgroup of | C2×A4⋊C16, #73104, #73105, … |
| maximal quotient of | C4×C8×A4, A4×C22⋊C8, A4×C4⋊C8, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(192, 1010);