Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D7 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4×C8 · Schur multiplier (H2) = C4 |
| direct product | C3×C4×C7⋊C8 |
| derived, Sylow | C7 ⋊ C3×C4×C8 |
| Hall | C3×C7 ⋊ C4×C8 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C42 | D7 |
| commutator, hypocenter | C7 | C3×C4×C8 |
| Frattini | C2×C4 | C2×C3×D7 |
| Fitting | C3×C42×C7 | C2 |
| socle, prime radical | C22×C3×C7 | C2×C4 |
| cosocle | C2×C4×C7 | C22×C3 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C42 | |
88 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×C7⋊C8, #13050859, #13050860, … |
| maximal normal subgroup of | C3⋊C8×C7⋊C8, #13050859, #13050860, … |
| maximal quotient of | C3×C8×C7⋊C8, C3×#89250, C3×C4×C7⋊C16, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 141);