Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C5 ⋊ 2−1+4 |
| Sylow | C5 ⋊ #1007 |
| Sylow | C3 ⋊ #4150 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C24 |
| Fitting | C2×C3×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D5 |
| hypocenter | C3×C5 | 2−1+4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,5,3)(6,7,8)(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24);(11,17)(14,21)(18,23)(22,24);(7,8)(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23);(2,3)(4,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)Polycyclic presentation:
2^5,3,5|a^2=e,d^2=e,d^a=de,g^a=g^4,c^b=ce,f^b=f^263 conjugacy classes — view the full character table.
| maximal subgroup of | #11747120, #11747121, #11748258, … |
| maximal normal subgroup of | #11747120, #11747121, #11748258, … |
| maximal quotient of | #11752460, #11752461, #11752464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1080);