Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#5930 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | #1020 |
| commutator, cosocle, hypocenter | C22⋊SL(2,𝔽3) | C2×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting | C22×C7×Q8 | D3 |
| socle | C23×C7 | S4 |
| prime radical | C7×C22⋊SL(2,𝔽3) | C2 |
| LCS | G ▷ C22⋊SL(2,𝔽3) | |
| UCS | C1 ◁ C2×C7 | |
128 conjugacy classes of subgroups; only the 18 normal subgroups are shown.
This group has 128 conjugacy classes of subgroups — too many to draw. Only its 18 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 336 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,2,3,4,5,6,7)(8,9,11,10)(12,17,20,27,16,25,22,18)(13,21,24,15,19,14,26,23);(9,10,11)(14,20,23)(15,21,22)(17,24,27)(18,25,26)105 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11416);