Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| direct product | D7×#41⋊C3 |
| Fitting | C7×#41 ⋊ C2×C3 |
| Hall | D7×#41 ⋊ C3 |
| Sylow | C7 ⋊ C2×#41⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | A4×D7 |
| commutator, cosocle, hypocenter | C7×Q8 | C22×C3 |
| Frattini | C2 | C2×A4×D7 |
| Fitting | C7×#41 | C2×C3 |
| socle | C2×C7 | C22×A4 |
| LCS | G ▷ C7×Q8 | |
| UCS | C1 ◁ C4 | |
98 conjugacy classes of subgroups.
| transitive degree | μt(G) = 112 |
70 conjugacy classes — view the full character table.
| maximal subgroup of | D7×GU(2,𝔽3), #13057087, D7×#131⋊C3, … |
| maximal normal subgroup of | D7×GU(2,𝔽3), #13057087, D7×#131⋊C3, … |
| maximal quotient of | C4×#66662, #13057582, C4×D7×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1111);