Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×#72215896 (order 18432) · Inn(G) = C22 · Out(G) = C2×#72215889 |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| direct product | C22×C3×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23×C3 | C22 |
| commutator, Frattini, cosocle | C2 | C24×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C3 ◁ G | |
156 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 15 points:
(1,2)(5,6,7);(3,4);(8,9,11,13)(10,14,15,12);(1,2)(3,4)(8,10,11,15)(9,12,13,14)60 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 222);