Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Hall | #90618 ⋊ C3 |
| Sylow | C7 ⋊ #5950 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×C22⋊A4 |
| commutator, Fitting, cosocle, hypocenter | C7×H(5,𝔽2) | C2×C3 |
| socle | C2×C7 | C2×C22⋊A4 |
| LCS | G ▷ C7×H(5,𝔽2) | |
| UCS | C1 ◁ C2 | |
330 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 330 conjugacy classes of subgroups — too many to draw. Only its 26 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,g^a=g^6,c^b=d,d^b=cd,e^b=f,f^b=ef,f^c=fh,e^d=eh55 conjugacy classes — view the full character table.
| maximal subgroup of | #72229375, D7×#5950, #72259583, … |
| maximal normal subgroup of | #72229375, D7×#5950, #72259583, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11494);