Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of ΓL(1,𝔽16) |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C16 · Schur multiplier (H2) = C2 |
| direct product | C2×C3⋊#674 |
| unambiguous semidirect product | (C2×C3)⋊#674 |
| derived, Hall | C3×C5 ⋊ C2×C16 |
| Sylow | C5 ⋊ C2×C3⋊C16 |
| Sylow | C3 ⋊ C2×#674 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2×C4 | ΓL(1,𝔽16) |
| commutator, hypocenter | C3×C5 | C2×C16 |
| Frattini | C4 | C2×ΓL(1,𝔽16) |
| Fitting | C2×C3×C4×C5 | C4 |
| socle, prime radical | C22×C3×C5 | C8 |
| cosocle | C3×#168 | C22 |
| LCS | G ▷ C3×C5 | |
| UCS | C1 ◁ C2×C4 | |
56 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 480 |
A minimal faithful permutation representation — 2 generators on 26 points:
(4,5)(7,8,9,10)(11,12,13,16,14,17,19,22,15,18,20,23,21,24,25,26);(1,2)(3,4,5)(6,7,10,8,9)72 conjugacy classes — view the full character table.
| maximal subgroup of | Dic3×#674, #11746541, #11746542, … |
| maximal normal subgroup of | Dic3×#674, #11746541, #11746542, … |
| maximal quotient of | C4×C3⋊#674, C4⋊(C3⋊#674), C2×C3⋊#3928, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 302);