Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | Aut(G) = C22×C4×S4 (order 384) · Inn(G) = C2×S4 · Out(G) = C2×C4 |
| homology | Abelianization (H1) = C22×C5 · Schur multiplier (H2) = C22 |
| direct product | C5×GL(2,ℤ/4) |
| Fitting | C24×C5 ⋊ D3 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C5 | C2×S4 |
| commutator, cosocle | C2×A4 | C22×C5 |
| Frattini | C2 | C2×C5×S4 |
| Fitting | C24×C5 | D3 |
| socle | C23×C5 | C2×D3 |
| hypercenter | C22×C5 | S4 |
| hypocenter | A4 | C5×D4 |
| LCS | G ▷ C2×A4 ▷ A4 | |
| UCS | C1 ◁ C2×C5 ◁ C22×C5 | |
124 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 24 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:
(2,4)(5,6,8)(9,10,11,12,13);(1,2)(3,4)(6,7)70 conjugacy classes — view the full character table.
| maximal subgroup of | #11756789, #11756794, D5×GL(2,ℤ/4), … |
| maximal normal subgroup of | #11756789, #11756794, D5×GL(2,ℤ/4), … |
| maximal quotient of | C5×SL(2,ℤ/4)⋊C4, C5×(C2×S4)⋊C4, C5×#5414, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1023);