Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of C7⋊#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | C7⋊#5936 |
| derived | C7×C22⋊A4 ⋊ C4 |
| Sylow | C7 ⋊ #5936 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | C7⋊#1020 |
| commutator, hypocenter | C7×C22⋊A4 | C4 |
| Fitting, socle | C25×C7 | D3 |
| prime radical, cosocle | C2×C7×C22⋊A4 | C2 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 ◁ C2 | |
300 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2^2,3,7,2^4|a^2=b,c^a=c^2,d^a=d^6,e^a=ef,g^a=gh,e^c=f,f^c=ef,g^c=h,h^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#5936, #72276617, #72276643, … |
| maximal normal subgroup of | D7×#5936, #72276617, #72276643, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11429);