Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Sylow | C7 ⋊ #984 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D7×S4 |
| commutator, cosocle, hypocenter | C7×SL(2,𝔽3) | C22 |
| Fitting | C7×Q8 | C2×D3 |
| socle | C2×C7 | C2×S4 |
| prime radical | #66662 | C2 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C2 | |
72 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,7,2^3|a^2=g,b^2=g,e^2=g,f^2=g,b^a=bg,d^a=d^6,c^b=c^2,e^b=fg,f^b=eg,e^c=f,f^c=efg,f^e=fg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#10621469, #13062043, #13062047, … |
| maximal normal subgroup of | C2×#10621469, #13062043, #13062047, … |
| maximal quotient of | #13057053, #13057054, #13057056, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 1051);