Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×D42 (order 384) · Inn(G) = C2×D12 · Out(G) = C23 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C3×M16 ⋊ C2 |
| Sylow | C3 ⋊ #137 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×D12 |
| commutator, cosocle | C3×C4 | C23 |
| Frattini | C4 | C22×D3 |
| Fitting | C3×M16 | C2 |
| socle | C2×C3 | C2×D4 |
| prime radical | #235 | C2 |
| hypercenter | M16 | D3 |
| hypocenter | C3 | #137 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ M16 | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 19 points:
(1,2,3)(4,7,8,15,9,16,6,13)(5,11,10,14,12,19,17,18);(4,6,9,8)(5,10,12,17)(7,15,16,13)(11,18,19,14);(2,3)(4,5)(6,10)(7,14)(8,17)(9,12)(11,15)(13,19)(16,18)Polycyclic presentation:
2^5,3|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^2,c^b=ce21 conjugacy classes — view the full character table.
| maximal subgroup of | #4746, #4748, #4752, … |
| maximal normal subgroup of | #4746, #4748, #4752, … |
| maximal quotient of | #4702, #4703, #4710, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 116);