Finite Group DatabasebetaFinite Group Databasebeta| order | 315 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) = C22×C32×C4 (order 144) · Inn(G) = C1 · Out(G) = C22×C32×C4 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C1 |
| cyclic | C315 |
| abelian | C5×C7×C9 |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C3 | C3×C5×C7 |
| socle, prime radical | C3×C5×C7 | C3 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 315 |
A minimal faithful permutation representation — 1 generator on 21 points:
(1,2,3,4,5,6,7,8,9)(10,11,12,13,14)(15,16,17,18,19,20,21)315 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(315, 2);