Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #5421 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D7×S4 |
| commutator, cosocle | C2×C7×SL(2,𝔽3) | C22 |
| Fitting | C2×C7×Q8 | C2×D3 |
| socle | C22×C7 | C2×S4 |
| hypocenter | C7×SL(2,𝔽3) | D4 |
| LCS | G ▷ C2×C7×SL(2,𝔽3) ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
190 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^3,3,7,2^3|f^2=h,g^2=h,b^a=bc,e^a=e^6,d^b=d^2,f^b=fg,g^b=gh,f^d=g,g^d=fgh,g^f=gh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6365);