Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ Q32 |
| Sylow | C7 ⋊ #829 |
| Sylow | C3 ⋊ Dic56 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #66561 |
| commutator, Fitting, cosocle | C3×C7×C8 | C22 |
| Frattini, hypercenter | C8 | D3×D7 |
| socle, prime radical | C2×C3×C7 | D8 |
| hypocenter | C3×C7 | Q32 |
| LCS | G ▷ C3×C7×C8 ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^5,3,7|a^2=e,b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,g^a=g^6,f^b=f^290 conjugacy classes — view the full character table.
| maximal subgroup of | D3×Dic56, #13052209, #13052214, … |
| maximal normal subgroup of | D3×Dic56, #13052209, #13052214, … |
| maximal quotient of | #13050926, #13050929, #13050969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 64);