Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C7 ⋊ #97 |
| Sylow | C7 ⋊ #815 |
| Sylow | C3 ⋊ #6377 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D3×D7 |
| commutator | C2×C3×C7 | C42 |
| Fitting, cosocle | C2×C3×C4×C7 | C22 |
| socle, prime radical | C22×C3×C7 | C2×C4 |
| hypocenter | C3×C7 | #97 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2×C4 | |
80 conjugacy classes of subgroups.
| transitive degree | μt(G) = 672 |
Polycyclic presentation:
2^5,3,7|a^2=c,b^2=d,c^2=e,b^a=be,g^a=g^6,f^b=f^2108 conjugacy classes — view the full character table.
| maximal subgroup of | #13052228, #13052232, #13052236, … |
| maximal normal subgroup of | #13052228, #13052232, #13052236, … |
| maximal quotient of | #13050859, #13050861, #13050905, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 70);