Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×C3×C4×C7 ⋊ C2 |
| Hall | C3×C7 ⋊ #121 |
| Sylow | C7 ⋊ #930 |
| Sylow | C3 ⋊ #6493 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C22×D21 |
| commutator, socle, cosocle | C22×C3×C7 | C23 |
| Fitting | C22×C3×C4×C7 | C2 |
| hypercenter | C22×C4 | D21 |
| hypocenter | C3×C7 | #121 |
| LCS | G ▷ C22×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C22 ◁ C22×C4 | |
188 conjugacy classes of subgroups; only the 63 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 63 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|b^2=e,b^a=be,c^a=cd,f^a=f^2,g^a=g^6174 conjugacy classes — view the full character table.
| maximal subgroup of | #13050986, #13050987, #13051355, … |
| maximal normal subgroup of | #13050986, #13050987, #13051355, … |
| maximal quotient of | #13056570, #13056576, #13056577, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 995);