Finite Group DatabasebetaFinite Group Databasebeta| order | 315 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C3×C4×AGL(1,𝔽7) (order 504) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×C3×C4 |
| homology | Abelianization (H1) = C5×C9 · Schur multiplier (H2) = C1 |
| direct product | C5×C7⋊C9 |
| unambiguous semidirect product | C7⋊(C5×C9) |
| unambiguous semidirect product | (C5×C7)⋊C9 |
| derived, Sylow | C7 ⋊ C5×C9 |
| Hall | C5×C7 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C5 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C5×C9 |
| Frattini | C3 | C5×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C3×C5×C7 | C3 |
| cosocle | C3×C7 | C3×C5 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C5 | |
12 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 315 |
A minimal faithful permutation representation — 2 generators on 21 points:
(2,3,4)(5,7,6)(8,9,11,10,12,14,13,15,16)(17,18,19,20,21);(1,2,3,5,4,6,7)75 conjugacy classes — view the full character table.
| maximal subgroup of | C5×D7⋊C9, D5×C7⋊C9, C5⋊(D7⋊C9), … |
| maximal normal subgroup of | C5×D7⋊C9, D5×C7⋊C9, C5⋊(D7⋊C9), … |
| maximal quotient of | C2×C5×C7⋊C9, C5×C7⋊C27, C3×C5×C7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(315, 1);