Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C9×ΓL(1,𝔽8) ⋊ D4 |
| Sylow | C7 ⋊ C3×#611 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | D9×AGL(1,𝔽7) |
| commutator, Fitting, cosocle | C2×C7×C9 | C22×C3 |
| Frattini | C2×C3 | D3×AGL(1,𝔽7) |
| socle | C2×C3×C7 | C2×C3×D3 |
| hypocenter | C7×C9 | C3×D4 |
| LCS | G ▷ C2×C7×C9 ▷ C7×C9 | |
| UCS | C1 ◁ C2 | |
116 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 116 conjugacy classes of subgroups — too many to draw. Only its 31 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 252 |
Polycyclic presentation:
2^2,3,2,3,7,3|e^3=g,b^a=bd,f^a=f^6,e^b=e^2g^2,g^b=g^2,f^c=f^275 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×#611, #72278028 |
| maximal normal subgroup of | AGL(1,𝔽7)×#611, #72278028 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 112);