Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of SL(2,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
| direct product | C22×SL(2,𝔽7) |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, socle, hypercenter | C23 | GL(3,𝔽2) |
| commutator, hypocenter | SL(2,𝔽7) | C22 |
| Frattini, cosocle | C2 | C22×GL(3,𝔽2) |
| LCS | G ▷ SL(2,𝔽7) | |
| UCS | C1 ◁ C23 | |
187 conjugacy classes of subgroups.
| transitive degree | μt(G) = 64 |
44 conjugacy classes — view the full character table.
| maximal subgroup of | A4×SL(2,𝔽7), D4×SL(2,𝔽7) |
| maximal normal subgroup of | A4×SL(2,𝔽7), D4×SL(2,𝔽7) |
| maximal quotient of | Q8×SL(2,𝔽7), D4×SL(2,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11305);