Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D4×S4 (order 192) · Inn(G) = C2×S4 · Out(G) = C22 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | A4⋊Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×S4 |
| commutator, prime radical, cosocle | C2×A4 | C22 |
| Fitting | C22×C4 | D3 |
| socle | C23 | C2×D3 |
| hypercenter | C4 | S4 |
| hypocenter | A4 | Q8 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2 ◁ C4 | |
42 conjugacy classes of subgroups.
| minimal degree | μ(G) = 12 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 2 generators on 12 points:
(1,2,4)(5,7,8,12)(6,9,10,11);(2,3)(5,6,8,10)(7,11,12,9)Polycyclic presentation:
2^3,3,2^2|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=f,f^a=e,e^d=f,f^d=ef14 conjugacy classes — view the full character table.
| maximal subgroup of | #5398, #5401, #5414, … |
| maximal normal subgroup of | #5398, #5401, #5414, … |
| maximal quotient of | #5386, #5389, #5391, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 185);