Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C7 · Schur multiplier (H2) = C2 |
| direct product | C7×#994 |
| Fitting, Hall | C7×H(5,𝔽2) ⋊ C3 |
| Sylow | H(5,𝔽2) ⋊ C3×C7 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C7 | C22×A4 |
| commutator, cosocle, hypocenter | Q8 | C22×C3×C7 |
| Frattini | C2 | C22×C7×A4 |
| Fitting | C7×H(5,𝔽2) | C3 |
| hypercenter | C7×Q8 | A4 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C2×C7 ◁ C7×Q8 | |
92 conjugacy classes of subgroups.
| transitive degree | μt(G) = 168 |
133 conjugacy classes — view the full character table.
| maximal subgroup of | C7×#5428, C7×#5429, #13057196, … |
| maximal normal subgroup of | C7×#5428, C7×#5429, #13057196, … |
| maximal quotient of | C7×#520⋊C3, C7×C4⋊(#41⋊C3), C7×Q8×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1190);