Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4×C7 · Schur multiplier (H2) = C23 |
| direct product | C2×C3×C7×#101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C3×C7 | C22⋊C4 |
| commutator | C22 | C22×C3×C4×C7 |
| Frattini, cosocle | C2×C4 | C23×C3×C7 |
| prime radical | C23×C3×C7 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C7 ◁ C22×C3×C4×C7 ◁ G | |
292 conjugacy classes of subgroups; only the 164 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 672 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(19,20,21)(22,23,24,25,26,27,28);(3,4,6,10,7,9,15,17)(5,11,12,18,13,8,14,16);(3,5,7,13)(4,8,9,11)(6,14,15,12)(10,18,17,16)462 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6212);