Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C5 ⋊ #135 |
| Sylow | C5 ⋊ #912 |
| Sylow | C3 ⋊ #4062 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6790 |
| commutator, cosocle | C3×C4×C5 | C23 |
| Frattini | C4 | C2×D3×D5 |
| Fitting | C3×C5×D4 | C22 |
| socle | C2×C3×C5 | C2×D4 |
| hypercenter | D4 | D3×D5 |
| hypocenter | C3×C5 | #135 |
| LCS | G ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ D4 | |
124 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 38 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)(10,15)(11,18)(12,21)(16,17)(19,20)(22,24);(7,8)(9,11,13,20)(10,14,17,23)(12,18,21,19)(15,22,16,24);(2,3)(4,5)(9,10)(11,14)(12,15)(13,17)(16,21)(18,22)(19,24)(20,23)Polycyclic presentation:
2^5,3,5|b^2=e,d^2=e,c^a=cd,d^a=de,g^a=g^4,c^b=cd,d^b=de,f^b=f^2,d^c=de45 conjugacy classes — view the full character table.
| maximal subgroup of | D5×#912, #11754049, #11754055, … |
| maximal normal subgroup of | D5×#912, #11754049, #11754055, … |
| maximal quotient of | #11747151, #11747184, #11747212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 573);