Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| automorphism | ? |
| homology | Abelianization (H1) = C4 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊#859 |
| derived | C7×SL(2,𝔽3) ⋊ C4 |
| Sylow | C7 ⋊ #859 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | C7⋊S4 |
| commutator, hypocenter | C7×SL(2,𝔽3) | C4 |
| Fitting | C2×C7×Q8 | D3 |
| socle | C22×C7 | S4 |
| prime radical, cosocle | C2×C7×SL(2,𝔽3) | C2 |
| LCS | G ▷ C7×SL(2,𝔽3) | |
| UCS | C1 ◁ C22 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 224 |
Polycyclic presentation:
2^2,3,7,2^3|a^2=b,e^2=g,f^2=g,c^a=c^2,d^a=d^6,e^a=fg,f^a=eg,e^c=f,f^c=efg,f^e=fg58 conjugacy classes — view the full character table.
| maximal subgroup of | #13057050, Dic7×#226, #13057053, … |
| maximal normal subgroup of | #13057050, Dic7×#226, #13057053, … |
| maximal quotient of | C7⋊(SL(2,𝔽3)⋊C8), C2×C7⋊#859, #72218633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 284);