Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of SO(3,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C2×GL(3,𝔽2)⋊C4 |
| unambiguous semidirect product | (C2×GL(3,𝔽2))⋊C4 |
| derived | GL(3,𝔽2) ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C22 | SO(3,𝔽7) |
| commutator, hypocenter | GL(3,𝔽2) | C2×C4 |
| Frattini | C2 | C2×SO(3,𝔽7) |
| socle, prime radical | C22×GL(3,𝔽2) | C2 |
| cosocle | C2×GL(3,𝔽2) | C22 |
| LCS | G ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C22 | |
149 conjugacy classes of subgroups; only the 13 normal subgroups are shown.
This group has 149 conjugacy classes of subgroups — too many to draw. Only its 13 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
36 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11294);