Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C7×#41 ⋊ C2×C3 |
| Hall | #90616 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #585 |
| Sylow | C7 ⋊ C3×#585 |
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C25×C3 |
| Frattini | C2 | C24×AGL(1,𝔽7) |
| Fitting | C2×C7×#41 | C2×C3 |
| hypercenter | C2×#41 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#585 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C4 ◁ C2×#41 | |
1620 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,3,2,7|c^2=g,e^a=eg,h^a=h^6,d^b=dg,h^f=h^4122 conjugacy classes — view the full character table.
| maximal subgroup of | #72229363, #72229450, #72229461, … |
| maximal normal subgroup of | #72229363, #72229450, #72229461, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11443);