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 ⋊ #916 |
| Sylow | C3 ⋊ #4062 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6777 |
| commutator, cosocle | C3×C4×C5 | C23 |
| Frattini | C4 | C2×D3×D5 |
| Fitting | C3×C5×C8 | C22 |
| socle | C2×C3×C5 | C2×D4 |
| hypercenter | C8 | D3×D5 |
| hypocenter | C3×C5 | #135 |
| LCS | G ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ C8 | |
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)(9,12,13,11,14,18,21,19)(10,15,16,20,17,23,22,24);(7,8)(9,11,14,19)(10,15,17,23)(12,21,18,13)(16,24,22,20);(2,3)(4,5)(9,10)(11,15)(12,20)(13,22)(14,17)(16,21)(18,24)(19,23)Polycyclic presentation:
2^5,3,5|b^2=e,c^2=d,d^2=e,c^a=cd,d^a=de,g^a=g^4,c^b=cd,d^b=de,f^b=f^251 conjugacy classes — view the full character table.
| maximal subgroup of | #11753978, #11753980, #11753982, … |
| maximal normal subgroup of | #11753978, #11753980, #11753982, … |
| maximal quotient of | #11747573, #11747685, #11747687, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 352);