Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×C22⋊A4 (order 1152) · Inn(G) = C22×D3 · Out(G) = C22×A4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C3×C42 ⋊ C2 |
| Sylow | C3 ⋊ #126 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator, socle, cosocle | C22×C3 | C23 |
| Fitting | C3×C42 | C2 |
| prime radical | C22×D3 | C22 |
| hypercenter | C42 | D3 |
| hypocenter | C3 | #126 |
| LCS | G ▷ C22×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C42 | |
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)(5,9)(8,11)(12,14,15,19)(13,17,16,18);(4,6,7,10)(5,8,9,11)(13,16)(17,18);(2,3)(4,5)(6,8)(7,9)(10,11)(12,13)(14,18)(15,16)(17,19)Polycyclic presentation:
2^5,3|b^2=d,c^2=e,b^a=be,c^a=cde,f^a=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#876, #5479, #5527, … |
| maximal normal subgroup of | C2×#876, #5479, #5527, … |
| maximal quotient of | #4658, C3⋊#401, #4675, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 83);