Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of C7⋊#857 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C7⋊#857 |
| Fitting | C42×C7 ⋊ D3 |
| derived | C7×C42⋊C3 ⋊ C2 |
| Sylow | C7 ⋊ #857 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C42⋊C3 | C2 |
| Frattini | C22 | C7⋊S4 |
| Fitting | C42×C7 | D3 |
| socle | C22×C7 | S4 |
| LCS | G ▷ C7×C42⋊C3 | |
| UCS | C1 | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3,7,2^4|d^2=f,e^2=fg,b^a=b^2,c^a=c^6,d^a=eg,e^a=dg,f^a=fg,d^b=ef,e^b=de,f^b=fg,g^b=f34 conjugacy classes — view the full character table.
| maximal subgroup of | D7×#857, #857⋊D7, C2×C7⋊#857, … |
| maximal normal subgroup of | D7×#857, #857⋊D7, C2×C7⋊#857, … |
| maximal quotient of | C7⋊#4621, C7⋊#4622, C7⋊#4623, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 282);