Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3×C4 · Schur multiplier (H2) = C210 |
| direct product | C24×ΓL(1,𝔽8)⋊C4 |
| derived, Sylow | C7 ⋊ C24×C3×C4 |
| Hall | C24×Dic7 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ C24×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C25 | AGL(1,𝔽7) |
| commutator, hypocenter | C7 | C24×C3×C4 |
| Frattini | C2 | C24×AGL(1,𝔽7) |
| Fitting, socle | C25×C7 | C2×C3 |
| prime radical | C25×ΓL(1,𝔽8) | C2 |
| cosocle | C2×C7 | C25×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C25 | |
2724 conjugacy classes of subgroups.
| transitive degree | μt(G) = 448 |
224 conjugacy classes — view the full character table.
| maximal subgroup of | C24×ΓL(1,𝔽8)⋊AGL(1,𝔽5), C24×C4×AGL(1,𝔽7), #120×ΓL(1,𝔽8)⋊C4, … |
| maximal normal subgroup of | C24×C4×AGL(1,𝔽7), #120×ΓL(1,𝔽8)⋊C4, #72218604 |
| maximal quotient of | C24×ΓL(1,𝔽8)⋊AGL(1,𝔽5) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11448);