Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×#2846 (order 768) · Inn(G) = C22×D3 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C3×D4 ⋊ C2 |
| Sylow | C3 ⋊ #120 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C23 |
| Fitting | C2×C3×D4 | C2 |
| hypercenter | C2×D4 | D3 |
| hypocenter | C3 | #120 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 3 generators on 11 points:
(1,2,3)(5,7)(9,11);(8,9)(10,11);(2,3)(4,5)(6,7)Polycyclic presentation:
2^5,3|c^a=cd,f^a=f^2,c^b=ce24 conjugacy classes — view the full character table.
| maximal subgroup of | #4471, #4475, #4741, … |
| maximal normal subgroup of | #4471, #4475, #4741, … |
| maximal quotient of | #4946, #4947, #4955, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 144);