Finite Group DatabasebetaFinite Group Databasebeta| order | 616 |
| isoclinism | isoclinic class of D77 |
| automorphism | ? |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C7⋊(C11⋊C8) |
| unambiguous semidirect product | C11⋊(C7⋊C8) |
| derived, Hall | C7×C11 ⋊ C8 |
| Sylow | C11 ⋊ C7⋊C8 |
| Sylow | C7 ⋊ C11⋊C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C4 | D77 |
| commutator, hypocenter | C7×C11 | C8 |
| Fitting, cosocle | C4×C7×C11 | C2 |
| socle, prime radical | C2×C7×C11 | C4 |
| LCS | G ▷ C7×C11 | |
| UCS | C1 ◁ C4 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 616 |
Polycyclic presentation:
2^3,7,11|a^2=b,b^2=c,d^a=d^6,e^a=e^10160 conjugacy classes — view the full character table.
| maximal subgroup of | D11×C7⋊C8, D7×C11⋊C8, #11927625, … |
| maximal normal subgroup of | D11×C7⋊C8, D7×C11⋊C8, #11927625, … |
| maximal quotient of | C7⋊(C11⋊C16), C2×C7⋊(C11⋊C8), C3×C7⋊(C11⋊C8), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(616, 3);