Finite Group DatabasebetaFinite Group Databasebeta| order | 672 |
| isoclinism | isoclinic class of D7⋊(C42⋊C3) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | D7⋊(C42⋊C3) |
| Fitting, derived | C42×C7 ⋊ C2×C3 |
| Hall | C42×D7 ⋊ C3 |
| Sylow | C7 ⋊ C2×C42⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | C42×C7 | C2×C3 |
| Frattini | C22 | D7⋊A4 |
| socle | C22×C7 | C2×A4 |
| LCS | G ▷ C42×C7 | |
| UCS | C1 | |
56 conjugacy classes of subgroups.
| transitive degree | μt(G) = 84 |
Polycyclic presentation:
2,3,2^2,7,2^2|c^2=f,d^2=g,e^a=e^6,c^b=df,d^b=cd,e^b=e^4,f^b=g,g^b=fg32 conjugacy classes — view the full character table.
| maximal subgroup of | C2×D7⋊(C42⋊C3), #13057646, #13057647, … |
| maximal normal subgroup of | C2×D7⋊(C42⋊C3), #13057646, #13057647, … |
| maximal quotient of | #13052165, D7⋊(#95⋊C3), Dic7⋊(C42⋊C3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(672, 382);