Finite Group DatabasebetaFinite Group Databasebeta| order | 441 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = C2×C32×AGL(1,𝔽7) (order 756) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C22×C32 |
| homology | Abelianization (H1) = C7×C9 · Schur multiplier (H2) = C1 |
| direct product | C7×C7⋊C9 |
| unambiguous semidirect product | C7⋊(C7×C9) |
| derived | C7 ⋊ C7×C9 |
| Sylow | C72 ⋊ C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3×C7 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C7×C9 |
| Frattini | C3 | C7×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C3×C72 | C3 |
| cosocle | C3×C7 | C3×C7 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C3×C7 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 23 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 23 points:
(1,2,3,4,5,6,7)(9,10,11)(12,14,13)(15,22,21,17,16,23,20,19,18);(8,9,10,12,11,13,14)105 conjugacy classes — view the full character table.
| maximal subgroup of | C7×D7⋊C9, D7×C7⋊C9, (C7⋊C9)⋊D7, … |
| maximal normal subgroup of | C7×D7⋊C9, D7×C7⋊C9, (C7⋊C9)⋊D7, … |
| maximal quotient of | C2×C7×C7⋊C9, C7×C7⋊C27, C3×C7×C7⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(441, 5);