Finite Group DatabasebetaFinite Group Databasebeta| order | 1512 |
| isoclinism | isoclinic class of C7⋊((C22⋊C27)⋊C2) |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | C7⋊((C22⋊C27)⋊C2) |
| unambiguous semidirect product | (C22⋊C27)⋊D7 |
| derived | C7×C22⋊C27 ⋊ C2 |
| Sylow | C7 ⋊ (C22⋊C27)⋊C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C22⋊C27 | C2 |
| Frattini | C9 | C7⋊S4 |
| Fitting | C22×C7×C9 | D3 |
| socle | C22×C3×C7 | D9 |
| LCS | G ▷ C7×C22⋊C27 | |
| UCS | C1 | |
50 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 378 |
Polycyclic presentation:
2,3,7,3^2,2^2|b^3=d,d^3=e,b^a=b^2d^2e^2,c^a=c^6,d^a=d^2e^2,e^a=e^2,f^a=fg,f^b=g,g^b=fg129 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1512, 85);