Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C7⋊#5930 |
| Sylow | C7 ⋊ #5930 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, prime radical, cosocle, hypocenter | C7×C22⋊SL(2,𝔽3) | C2 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C23×C7 | S4 |
| LCS | G ▷ C7×C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
128 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2,3,7,2^5|a^2=h,f^2=h,g^2=h,b^a=b^2,c^a=c^6,d^a=de,f^a=fg,g^a=gh,d^b=e,e^b=de,f^b=g,g^b=fgh,g^f=gh51 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11423);