Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of D3×C3⋊AGL(1,𝔽7) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | Dic3×C3⋊AGL(1,𝔽7) |
| unambiguous semidirect product | (Dic3×D21)⋊C3 |
| derived | C32×C7 ⋊ C2×C3×C4 |
| Hall | C32×ΓL(1,𝔽8) ⋊ C2×C4 |
| Sylow | C7 ⋊ C3×D3×Dic3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×C3⋊AGL(1,𝔽7) |
| commutator, hypocenter | C32×C7 | C2×C3×C4 |
| Fitting, socle, cosocle | C2×C32×C7 | C22×C3 |
| prime radical | C2×C3×C3⋊AGL(1,𝔽7) | C2 |
| LCS | G ▷ C32×C7 | |
| UCS | C1 ◁ C2 | |
180 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 168 |
72 conjugacy classes — view the full character table.
| maximal subgroup of | C4×D3×C3⋊AGL(1,𝔽7), #72217523, D3×Dic3×AGL(1,𝔽7), … |
| maximal normal subgroup of | C4×D3×C3⋊AGL(1,𝔽7), #72217523, D3×Dic3×AGL(1,𝔽7), … |
| maximal quotient of | ΓL(1,𝔽16)×C3⋊AGL(1,𝔽7) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 620);