Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of D5×A4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×D5×A4 |
| unambiguous semidirect product | (C22×C4×D5)⋊C3 |
| Fitting | C22×C4×C5 ⋊ C2×C3 |
| derived | C22×C5 ⋊ C2×C3×C4 |
| Hall | C22×C4×D5 ⋊ C3 |
| Sylow | C5 ⋊ C2×C4×A4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D5×A4 |
| commutator, hypocenter | C22×C5 | C2×C3×C4 |
| Frattini | C2 | C2×D5×A4 |
| Fitting | C22×C4×C5 | C2×C3 |
| socle, cosocle | C23×C5 | C22×C3 |
| prime radical | C2×D5×A4 | C2 |
| LCS | G ▷ C22×C5 | |
| UCS | C1 ◁ C4 | |
132 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 132 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 60 |
A minimal faithful permutation representation — 2 generators on 13 points:
(6,7,8)(10,11)(12,13);(1,2,3,4)(5,8)(6,7)(9,10,12,13,11)64 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4⋊C8, #11751578, C8×D5×A4, … |
| maximal normal subgroup of | D5×A4⋊C8, #11751578, C8×D5×A4, … |
| maximal quotient of | C4×C5⋊(#41⋊C3), #65908⋊C3, C4×D5×SL(2,𝔽3), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1036);