Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×#2846 (order 768) · Inn(G) = C22×D3 · Out(G) = #120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C22×C3×C4 ⋊ C2 |
| Sylow | C3 ⋊ #123 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C23 |
| Fitting | C22×C3×C4 | C2 |
| prime radical | C2×#67 | C2 |
| hypercenter | C22×C4 | D3 |
| hypocenter | C3 | #123 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
78 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 48 |
A minimal faithful permutation representation — 3 generators on 15 points:
(1,2,3)(5,7)(8,10,11,14)(9,13,12,15);(9,12)(13,15);(2,3)(4,5)(6,7)(8,9)(10,13)(11,12)(14,15)Polycyclic presentation:
2^5,3|c^2=e,b^a=be,c^a=cd,f^a=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | #5053, #5479, #5490, … |
| maximal normal subgroup of | #5053, #5479, #5490, … |
| maximal quotient of | #4933, #4940, #4952, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 136);