Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| isoclinism | isoclinic class of SO(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2 |
| direct product | C3×SO(3,𝔽7) |
| unambiguous semidirect product | GL(3,𝔽2)⋊(C2×C3) |
| socle | C3×GL(3,𝔽2) ⋊ C2 |
| derived | GL(3,𝔽2) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C3 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C2×C3 |
| Frattini | C1 | G |
| socle | C3×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C3 | |
53 conjugacy classes of subgroups.
| transitive degree | μt(G) = 24 |
27 conjugacy classes — view the full character table.
| maximal subgroup of | C2×C3×SO(3,𝔽7), #72244479, D3×SO(3,𝔽7), … |
| maximal normal subgroup of | C2×C3×SO(3,𝔽7), D3×SO(3,𝔽7), C32×SO(3,𝔽7) |
| maximal quotient of | C3×#10621462, C2×C3×SO(3,𝔽7), #72244479, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 882);