Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = C22×D3×#120 (order 768) · Inn(G) = C22×D3 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Fitting | C3×C4⋊C4 ⋊ C2 |
| Sylow | C3 ⋊ #117 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D3 |
| commutator | C2×C3 | C22×C4 |
| Fitting | C3×C4⋊C4 | C2 |
| socle, cosocle | C22×C3 | C23 |
| prime radical | C22×D3 | C22 |
| hypercenter | C4⋊C4 | D3 |
| hypocenter | C3 | #117 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C22 ◁ C4⋊C4 | |
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)(4,5,6,7)(9,13)(10,14);(8,10,11,14)(9,12,13,15);(2,3)(8,9)(10,12)(11,13)(14,15)Polycyclic presentation:
2^5,3|b^2=e,c^2=d,c^a=ce,f^a=f^2,c^b=ce30 conjugacy classes — view the full character table.
| maximal subgroup of | #4770, #4772, #4779, … |
| maximal normal subgroup of | #4770, #4772, #4779, … |
| maximal quotient of | #4648, #4651, #4655, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 99);