Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | Q8×GL(3,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2 | C22×GL(3,𝔽2) |
| commutator, socle, prime radical | C2×GL(3,𝔽2) | C22 |
| Fitting, hypercenter | Q8 | GL(3,𝔽2) |
| hypocenter | GL(3,𝔽2) | Q8 |
| LCS | G ▷ C2×GL(3,𝔽2) ▷ GL(3,𝔽2) | |
| UCS | C1 ◁ C2 ◁ Q8 | |
171 conjugacy classes of subgroups.
| transitive degree | μt(G) = 56 |
30 conjugacy classes — view the full character table.
| maximal subgroup of | SL(2,𝔽3)×GL(3,𝔽2), SD16×GL(3,𝔽2), Q16×GL(3,𝔽2) |
| maximal normal subgroup of | SL(2,𝔽3)×GL(3,𝔽2), SD16×GL(3,𝔽2), Q16×GL(3,𝔽2) |
| maximal quotient of | Q8×SL(2,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11303);