Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of D48 |
| automorphism | Aut(G) = D3×#442 (order 384) · Inn(G) = D24 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C3×C16 ⋊ C2 |
| Sylow | C3 ⋊ SD32 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | D24 |
| commutator, cosocle | C3×C8 | C22 |
| Frattini | C8 | C2×D3 |
| Fitting | C3×C16 | C2 |
| socle | C2×C3 | D8 |
| prime radical | D24 | C2 |
| hypercenter | C16 | D3 |
| hypocenter | C3 | SD32 |
| LCS | G ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 ◁ C16 | |
26 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3)(4,5,10,16,15,19,6,9,8,12,13,18,7,11,14,17);(2,3)(5,9)(6,13)(7,15)(10,14)(11,18)(12,17)(16,19)Polycyclic presentation:
2^5,3|b^2=cd,c^2=d,d^2=e,b^a=bc,c^a=cde,d^a=de,f^a=f^227 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#800, #4904, #4908, … |
| maximal normal subgroup of | C2×#800, #4904, #4908, … |
| maximal quotient of | #4503, #4505, #4509, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 7);