Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D7×C42⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C4 |
| direct product | D7×C42⋊C3 |
| Fitting, derived | C42×C7 ⋊ C2×C3 |
| Hall | C42×D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×C42⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C3 |
| Frattini | C22 | A4×D7 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
40 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#857, C2×D7×C42⋊C3, D7×#126⋊C3, … |
| maximal normal subgroup of | D7×#857, C2×D7×C42⋊C3, D7×#126⋊C3, … |
| maximal quotient of | #13052148, D7×#95⋊C3, Dic7×C42⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 374);