Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×#23900 (order 1536) · Inn(G) = C22×D3 · Out(G) = C2×#120 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C42 ⋊ C2 |
| Sylow | C3 ⋊ #124 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C23 |
| Fitting | C3×C42 | C2 |
| prime radical | C2×D12 | C2 |
| hypercenter | C42 | D3 |
| hypocenter | C3 | #124 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 | |
76 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)(4,6,7,5)(9,13)(12,15);(4,5,7,6)(8,10,11,14)(9,12,13,15);(2,3)(5,6)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^5,3|b^2=d,c^2=e,b^a=be,c^a=cd,f^a=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | #4464, #4696, #4697, … |
| maximal normal subgroup of | #4464, #4696, #4697, … |
| maximal quotient of | #4659, #4672, #4673, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 82);