Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C7 ⋊ D16 |
| Sylow | C7 ⋊ D48 |
| Sylow | C3 ⋊ #6400 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #66562 |
| commutator, Fitting, cosocle | C3×C7×C8 | C22 |
| Frattini, hypercenter | C8 | D3×D7 |
| socle | C2×C3×C7 | D8 |
| hypocenter | C3×C7 | D16 |
| LCS | G ▷ C3×C7×C8 ▷ C3×C4×C7 ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 336 |
Polycyclic presentation:
2^5,3,7|a^2=c,c^2=de,d^2=e,b^a=bc,g^a=g^6,c^b=cd,d^b=de,f^b=f^272 conjugacy classes — view the full character table.
| maximal subgroup of | D7×D48, #13052195, #13052196, … |
| maximal normal subgroup of | D7×D48, #13052195, #13052196, … |
| maximal quotient of | #13050848, #13050853, #13050854, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 56);