Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of C3⋊(A4⋊AGL(1,𝔽7)) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | C3⋊(A4⋊AGL(1,𝔽7)) |
| unambiguous semidirect product | A4⋊(C3⋊AGL(1,𝔽7)) |
| unambiguous semidirect product | ΓL(1,𝔽8)⋊(C3⋊S4) |
| Fitting, socle | C22×C3×C7 ⋊ C3×D3 |
| derived | C3×C7×A4 ⋊ C2×C3 |
| Sylow | C7 ⋊ C3×C3⋊S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C7×A4 | C2×C3 |
| Fitting, socle | C22×C3×C7 | C3×D3 |
| LCS | G ▷ C3×C7×A4 | |
| UCS | C1 | |
154 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 154 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.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3^3,7,2^2|c^a=c^2,d^a=d^2,e^a=e^6,f^a=fg,e^b=e^4,f^c=g,g^c=fg39 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 797);