Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Hall | C3×C7 ⋊ M32 |
| Sylow | C7 ⋊ #798 |
| Sylow | C3 ⋊ #6386 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C8 | D3×D7 |
| commutator, socle | C2×C3×C7 | C2×C8 |
| Fitting, cosocle | C3×C7×C8 | C22 |
| prime radical | C2×D3×C7 | C8 |
| hypocenter | C3×C7 | M32 |
| LCS | G ▷ C2×C3×C7 ▷ C3×C7 | |
| UCS | C1 ◁ C8 | |
52 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=d,d^2=e,b^a=be,g^a=g^6,f^b=f^2108 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#798, #13052188, #13052189, … |
| maximal normal subgroup of | D7×#798, #13052188, #13052189, … |
| maximal quotient of | #13050902, (C2×D3)⋊(C7⋊C16), Dic3⋊(C7⋊C16), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 52);