Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting, Hall | C7×#585 ⋊ C3 |
| Sylow | #585 ⋊ ΓL(1,𝔽8) |
| Sylow | C7 ⋊ #5950 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | C22⋊(C22⋊ΓL(1,𝔽8)) |
| commutator, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C3 |
| Frattini | C2 | C2×C22⋊(C22⋊ΓL(1,𝔽8)) |
| Fitting | C7×#585 | C3 |
| socle | C2×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C4 | |
330 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 32 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:
2,3,2^4,7,2|a^2=h,c^b=cd,d^b=c,e^b=ef,f^b=e,g^b=g^2,f^c=fh,e^d=eh90 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#5950, #72230067, #72230198, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#5950, #72230067, #72230198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11616);