Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C7 · Schur multiplier (H2) = C22 |
| direct product | C7×#5933 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C2×C7 | #1020 |
| commutator, cosocle, hypocenter | #997 | C2×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting | C7×H(5,𝔽2) | D3 |
| prime radical | C7×#997 | C2 |
| LCS | G ▷ #997 | |
| UCS | C1 ◁ C2×C7 | |
132 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 16 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 112 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(8,21,16,22,12,13,19,14)(9,10,20,11,15,17,23,18);(10,16,18)(11,17,19)(13,20,22)(14,21,23)91 conjugacy classes — view the full character table.
| maximal subgroup of | ΓL(1,𝔽8)×#5933, C7×#87520, C7×#87516, … |
| maximal normal subgroup of | ΓL(1,𝔽8)×#5933, C7×#87520, C7×#87516, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11419);