Finite Group DatabasebetaFinite Group Databasebeta| order | 405 |
| isoclinism | isoclinic class of C34⋊C5 |
| automorphism | ? |
| homology | Abelianization (H1) = C5 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | C34⋊C5 |
| Fitting, socle, derived, Sylow | C34 ⋊ C5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, Fitting, socle, cosocle, hypocenter | C34 | C5 |
| LCS | G ▷ C34 | |
| UCS | C1 | |
46 conjugacy classes of subgroups.
| minimal degree | μ(G) = 15 |
| transitive degree | μt(G) = 15 |
A minimal faithful permutation representation — 2 generators on 15 points:
(1,2,4,8,11)(3,6,10,12,15)(5,9,13,14,7);(1,3,7)(2,6,5)(8,13,12)(11,14,15)Polycyclic presentation:
5,3^4|b^a=cd^2e^2,c^a=b^2d,d^a=c^2e,e^a=bd^2e^221 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(405, 15);