Finite Group DatabasebetaFinite Group Databasebeta| order | 1008 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #66694⋊C3 |
| Fitting | C22×C3×C7 ⋊ C22×C3 |
| Hall | C3×ΓL(1,𝔽8) ⋊ C2×D4 |
| Sylow | C7 ⋊ C3×D3×D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×AGL(1,𝔽7) |
| commutator, socle, cosocle | C2×C3×C7 | C23×C3 |
| Fitting | C22×C3×C7 | C22×C3 |
| hypercenter | C22 | D3×AGL(1,𝔽7) |
| hypocenter | C3×C7 | C2×C3×D4 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C22 | |
232 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 232 conjugacy classes of subgroups — too many to draw. Only its 59 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:
2^3,3,2,3,7|c^a=ce,g^a=g^6,c^b=ce,f^b=f^2,g^d=g^254 conjugacy classes — view the full character table.
| maximal subgroup of | #72222036, #72222040, D3×D4×AGL(1,𝔽7), … |
| maximal normal subgroup of | D3×D4×AGL(1,𝔽7), C2×#66694⋊C3, #72244406, … |
| maximal quotient of | #72222036, #72222040, #72222098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1008, 553);