Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C5 ⋊ #125 |
| Sylow | C5 ⋊ #890 |
| Sylow | C3 ⋊ #4035 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D5 |
| commutator, socle, prime radical, cosocle | C22×C3×C5 | C23 |
| Fitting | C2×C3×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D5 |
| hypocenter | C3×C5 | #125 |
| LCS | G ▷ C22×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
112 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 112 conjugacy classes of subgroups — too many to draw. Only its 48 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) = 480 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,5,3)(6,7,8)(10,14)(11,15)(17,20,21,19)(18,22,23,24);(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,20)(18,22,23,24);(2,3)(4,5)(9,10,12,14)(11,13,15,16)(17,18,21,23)(19,22,20,24)Polycyclic presentation:
2^5,3,5|a^2=d,b^2=de,c^2=de,c^a=cd,g^a=g^4,c^b=ce,f^b=f^260 conjugacy classes — view the full character table.
| maximal subgroup of | #11752469, #11752476, #11752488, … |
| maximal normal subgroup of | #11752469, #11752476, #11752488, … |
| maximal quotient of | #11748568, #11748570, #11748581, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 410);