Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| automorphism | Aut(G) = D3×D4×S4 (order 1152) · Inn(G) = C23×D3 · Out(G) = C22×D3 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Fitting | C3×#41 ⋊ C2 |
| Sylow | C3 ⋊ H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3 |
| commutator, socle, cosocle | C2×C3 | C24 |
| Fitting | C3×#41 | C2 |
| hypercenter | #41 | D3 |
| hypocenter | C3 | H(5,𝔽2) |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2 ◁ #41 | |
166 conjugacy classes of subgroups.
| minimal degree | μ(G) = 11 |
| transitive degree | μt(G) = 24 |
A minimal faithful permutation representation — 4 generators on 11 points:
(1,2,3)(4,6,7,10)(5,9,8,11);(5,8)(9,11);(4,5)(6,9)(7,8)(10,11);(2,3)(6,10)(9,11)Polycyclic presentation:
2^5,3|d^2=e,d^a=de,f^a=f^2,c^b=ce27 conjugacy classes — view the full character table.
| maximal subgroup of | #4822, #4825, #5198, … |
| maximal normal subgroup of | #4822, #4825, #5198, … |
| maximal quotient of | #5519, #5521, #5523, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 216);