Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C22⋊SO(3,𝔽7) |
| Fitting | C22 ⋊ SO(3,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×SO(3,𝔽7) |
| commutator, socle, cosocle | C2×GL(3,𝔽2) | C22 |
| Fitting, hypercenter | C22 | SO(3,𝔽7) |
| hypocenter | GL(3,𝔽2) | D4 |
| LCS | G ▷ C2×GL(3,𝔽2) ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 ◁ C22 | |
147 conjugacy classes of subgroups; only the 9 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 9 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 28 |
Permutation generators:
(1,3,4)(5,6,8)(11,12);(1,3,2,6,4,7,8,5)(9,12)(10,11)24 conjugacy classes — view the full character table.
| maximal quotient of | #72244484 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11295);