Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| isoclinism | isoclinic class of #1020 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C7×#5936 |
| derived | C22⋊A4 ⋊ C4×C7 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C7 | #1020 |
| commutator, hypocenter | C22⋊A4 | C4×C7 |
| Frattini | C2 | C7×#1020 |
| Fitting, socle | C25×C7 | D3 |
| prime radical | C2×C7×C22⋊A4 | C2 |
| cosocle | C2×C22⋊A4 | C2×C7 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C2×C7 | |
300 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 300 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
A minimal faithful permutation representation — 2 generators on 19 points:
(1,2,3,4,5,6,7)(8,9,10,12)(11,19)(13,18)(14,15)(16,17);(8,11,16)(9,13,15)(10,14,18)(12,17,19)140 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#5936, ΓL(1,𝔽8)×#5936, #72276689, … |
| maximal normal subgroup of | D7×#5936, ΓL(1,𝔽8)×#5936, #72276689, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 11422);