Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5 ⋊ C22⋊Q8 |
| Sylow | C5 ⋊ #924 |
| Sylow | C3 ⋊ #4024 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D5 |
| commutator, socle, cosocle | C22×C3×C5 | C23 |
| Fitting, prime radical | C23×C3×C5 | C22 |
| hypercenter | C23 | D3×D5 |
| hypocenter | C3×C5 | C22⋊Q8 |
| LCS | G ▷ C22×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C23 | |
148 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 148 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 240 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,5,3)(6,7,8)(10,12);(7,8)(9,10)(11,12)(13,15,16,20)(14,17,18,19);(2,3)(4,5)(13,14,16,18)(15,19,20,17)Polycyclic presentation:
2^5,3,5|a^2=d,b^2=d,b^a=bd,g^a=g^4,c^b=ce,f^b=f^266 conjugacy classes — view the full character table.
| maximal subgroup of | #11752778, #11752779, #11752780, … |
| maximal normal subgroup of | #11752778, #11752779, #11752780, … |
| maximal quotient of | #11748802, #11748815, #11748816, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 650);