Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D21 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C3⋊(C7⋊C32) |
| unambiguous semidirect product | C7⋊(C3⋊C32) |
| derived, Hall | C3×C7 ⋊ C32 |
| Sylow | C7 ⋊ C3⋊C32 |
| Sylow | C3 ⋊ C7⋊C32 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C16 | D21 |
| commutator, hypocenter | C3×C7 | C32 |
| Fitting, cosocle | C3×C7×C16 | C2 |
| socle, prime radical | C2×C3×C7 | C16 |
| LCS | G ▷ C3×C7 | |
| UCS | C1 ◁ C16 | |
24 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^5,3,7|a^2=b,b^2=c,c^2=d,d^2=e,f^a=f^2,g^a=g^6192 conjugacy classes — view the full character table.
| maximal subgroup of | D7×C3⋊C32, D3×C7⋊C32, #13050843, … |
| maximal normal subgroup of | D7×C3⋊C32, D3×C7⋊C32, #13050843, … |
| maximal quotient of | C3⋊(C7⋊C64), C2×C3⋊(C7⋊C32) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 5);