Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of D7×#1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C24 |
| direct product | D7×#1020 |
| Fitting, socle | C24×C7 ⋊ C2×D3 |
| derived | C7×C22⋊A4 ⋊ C22 |
| Sylow | C7 ⋊ C2×#1020 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C22⋊A4 | C22 |
| Fitting, socle | C24×C7 | C2×D3 |
| LCS | G ▷ C7×C22⋊A4 | |
| UCS | C1 | |
476 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 476 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 56 |
50 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#1020, D7×#1020⋊C2, #1020×C7⋊D7, … |
| maximal normal subgroup of | AGL(1,𝔽7)×#1020, D7×#1020⋊C2, C2×D7×#1020 |
| maximal quotient of | D7×#5935, D7×#5934, #1020×C7⋊D7, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11696);