Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of SO(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| general orthogonal | GO(3,𝔽7) |
| direct product | C2×SO(3,𝔽7) |
| unambiguous semidirect product | GL(3,𝔽2)⋊C22 |
| unambiguous semidirect product | (C2×GL(3,𝔽2))⋊C2 |
| socle | C2×GL(3,𝔽2) ⋊ C2 |
| derived | GL(3,𝔽2) ⋊ C22 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C2 | SO(3,𝔽7) |
| commutator, cosocle, hypocenter | GL(3,𝔽2) | C22 |
| Frattini | C1 | G |
| socle | C2×GL(3,𝔽2) | C2 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 | |
70 conjugacy classes of subgroups.
| transitive degree | μt(G) = 16 |
18 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1254);