Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C7 ⋊ #123 |
| Sylow | C7 ⋊ #886 |
| Sylow | C3 ⋊ #6498 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D7 |
| commutator, socle, cosocle | C22×C3×C7 | C23 |
| Fitting | C23×C3×C7 | C22 |
| prime radical | C2×C7×#67 | C2 |
| hypercenter | C23 | D3×D7 |
| hypocenter | C3×C7 | #123 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C23 | |
156 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 52 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|a^2=d,b^2=e,b^a=bd,c^a=ce,g^a=g^6,c^b=ce,f^b=f^278 conjugacy classes — view the full character table.
| maximal subgroup of | #13052515, #13054428, #13058056, … |
| maximal normal subgroup of | #13052515, #13054428, #13058056, … |
| maximal quotient of | #13053709, #13053841, #13053842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 687);