Finite Group DatabasebetaFinite Group Databasebeta| order | 1344 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| Fitting | C7×#120 ⋊ C2×C3 |
| Hall | #90285 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ H(3,𝔽4) |
| Sylow | C7 ⋊ C3×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C24×C3 |
| Fitting | C7×#120 | C2×C3 |
| hypercenter | #120 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×H(3,𝔽4) |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ #120 | |
520 conjugacy classes of subgroups; only the 162 normal subgroups are shown.
This group has 520 conjugacy classes of subgroups — too many to draw. Only its 162 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 112 |
Polycyclic presentation:
2^4,3,2^2,7|c^a=cg,d^a=dfg,h^a=h^6,c^b=cf,d^b=dg,h^e=h^271 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×H(3,𝔽4), #72276718, #72276750, … |
| maximal normal subgroup of | AGL(1,𝔽7)×H(3,𝔽4), #72276718, #72276750, … |
| maximal quotient of | #72276674 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1344, 6617);