Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D4×S4 (order 192) · Inn(G) = C22×A4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| direct product | D4×A4 |
| unambiguous semidirect product | C4⋊(C2×A4) |
| unambiguous semidirect product | (C22×D4)⋊C3 |
| Fitting, Sylow | C22×D4 ⋊ C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×A4 |
| commutator, socle, cosocle | C23 | C22×C3 |
| Fitting | C22×D4 | C3 |
| hypercenter | D4 | A4 |
| hypocenter | C22 | C3×D4 |
| LCS | G ▷ C23 ▷ C22 | |
| UCS | C1 ◁ C2 ◁ D4 | |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 8 |
| transitive degree | μt(G) = 12 |
A minimal faithful permutation representation — 2 generators on 8 points:
(2,4)(6,7,8);(1,2)(3,4)(5,8)(6,7)20 conjugacy classes — view the full character table.
| maximal subgroup of | #5414, #5415, A4×D8, … |
| maximal normal subgroup of | #5414, #5415, A4×D8, … |
| maximal quotient of | A4×C22⋊C4, A4×C4⋊C4, #548⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 197);