Finite Group DatabasebetaFinite Group Databasebeta| order | 96 |
| isoclinism | isoclinic class of #204 |
| automorphism | Aut(G) = C2×D3×AGL(1,ℤ/8) (order 384) · Inn(G) = D12 · Out(G) = C24 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | C2×C3×C8 ⋊ C2 |
| Sylow | C3 ⋊ #135 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | D12 |
| commutator, cosocle | C3×C4 | C23 |
| Frattini | C4 | C22×D3 |
| Fitting | C2×C3×C8 | C2 |
| socle | C2×C3 | C2×D4 |
| hypercenter | C2×C8 | D3 |
| hypocenter | C3 | #135 |
| LCS | G ▷ C3×C4 ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C2×C8 | |
62 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)(4,7,8,15,9,16,6,13)(5,11,10,14,12,18,17,19);(4,6,9,8)(5,10,12,17)(7,13,16,15)(11,14,18,19);(2,3)(4,5)(6,10)(7,14)(8,17)(9,12)(11,15)(13,18)(16,19)Polycyclic presentation:
2^5,3|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,f^a=f^230 conjugacy classes — view the full character table.
| maximal subgroup of | #4488, #4497, #4510, … |
| maximal normal subgroup of | #4488, #4497, #4510, … |
| maximal quotient of | #4683, C4×#204, C4×D24, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(96, 111);