Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | Dic7⋊(C3⋊C8) |
| Hall | C3×C7 ⋊ C4⋊C8 |
| Sylow | C7 ⋊ C4⋊(C3⋊C8) |
| Sylow | C3 ⋊ Dic7⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×D7 |
| commutator | C2×C3×C7 | C2×C8 |
| Fitting, cosocle | C2×C3×C4×C7 | C22 |
| socle, prime radical | C22×C3×C7 | C2×C4 |
| hypocenter | C3×C7 | C4⋊C8 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C4 | |
76 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^5,3,7|a^2=d,b^2=c,c^2=e,b^a=bd,g^a=g^6,f^b=f^2108 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊C8×Dic14, #13052224, #13052225, … |
| maximal normal subgroup of | C3⋊C8×Dic14, #13052224, #13052225, … |
| maximal quotient of | #13050892, #13050895, Dic7⋊(C3⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 99);