Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5 ⋊ #131 |
| Sylow | C5 ⋊ #901 |
| Sylow | C3 ⋊ #4053 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C4 | C2×D3×D5 |
| commutator, socle | C2×C3×C5 | C22×C4 |
| Fitting | C3×C5×C8 | C22 |
| prime radical | #6771 | C2 |
| cosocle | C3×C4×C5 | C23 |
| hypercenter | C8 | D3×D5 |
| hypocenter | C3×C5 | #131 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C4 ◁ C8 | |
124 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 50 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 — 3 generators on 24 points:
(1,2,4,5,3)(6,7,8)(9,11,12,18,13,17,19,23)(10,15,16,22,14,20,21,24);(7,8)(10,14)(11,17)(16,21)(18,23);(2,3)(4,5)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)Polycyclic presentation:
2^5,3,5|c^2=d,d^2=e,b^a=be,g^a=g^4,c^b=ce,f^b=f^272 conjugacy classes — view the full character table.
| maximal subgroup of | #11751642, #11751644, D5×#901, … |
| maximal normal subgroup of | #11751642, #11751644, D5×#901, … |
| maximal quotient of | #11747564, #11747569, #11747646, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 342);