Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #6502⋊C3 |
| Fitting | C2×C7×D4 ⋊ C2×C3 |
| Hall | #6502 ⋊ C3 |
| Hall | ΓL(1,𝔽8) ⋊ #121 |
| Sylow | C7 ⋊ C3×#121 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×AGL(1,𝔽7) |
| commutator, socle, cosocle | C22×C7 | C23×C3 |
| Fitting | C2×C7×D4 | C2×C3 |
| hypercenter | C2×D4 | AGL(1,𝔽7) |
| hypocenter | C7 | C3×#121 |
| LCS | G ▷ C22×C7 ▷ C7 | |
| UCS | C1 ◁ C22 ◁ C2×D4 | |
188 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 188 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) = 112 |
Polycyclic presentation:
2^3,3,2^2,7|b^a=bf,c^a=ce,g^a=g^6,c^b=cf,g^d=g^452 conjugacy classes — view the full character table.
| maximal subgroup of | #90243⋊C3, #90229⋊C3, #90231⋊C3, … |
| maximal normal subgroup of | #90243⋊C3, #90229⋊C3, #90231⋊C3, … |
| maximal quotient of | #89714⋊C3, #89729⋊C3, #89724⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 357);