Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C7×2−1+4 ⋊ C2×C3 |
| Hall | #90620 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #585 |
| Sylow | C7 ⋊ C3×#585 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C24×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C7 | C25×C3 |
| Fitting | C7×2−1+4 | C2×C3 |
| hypercenter | 2−1+4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#585 |
| LCS | G ▷ C2×C7 ▷ C7 | |
| UCS | C1 ◁ C2 ◁ 2−1+4 | |
1620 conjugacy classes of subgroups; only the 818 normal subgroups are shown.
This group has 1620 conjugacy classes of subgroups — too many to draw. Only its 818 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^5,3,2,7|c^2=g,e^2=g,e^a=eg,h^a=h^6,d^b=dg,e^c=eg,h^f=h^4119 conjugacy classes — view the full character table.
| maximal subgroup of | #72229639, AGL(1,𝔽7)×#585, #72230256, … |
| maximal normal subgroup of | #72229639, AGL(1,𝔽7)×#585, #72230256, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11447);