Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C7 ⋊ #5927 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D7×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C23 |
| Fitting | C2×C7×Q8 | C2×D3 |
| socle | C2×C7 | C22×S4 |
| hypercenter | C22 | D7×S4 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 ◁ C22 | |
280 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,7,2^3|b^2=h,f^2=h,g^2=h,c^a=ch,e^a=e^6,c^b=ch,d^b=d^2,f^b=fg,g^b=gh,f^d=g,g^d=fgh,g^f=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11337);