Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Hall | C3×C5 ⋊ 2−1+4 |
| Sylow | C5 ⋊ #1010 |
| Sylow | C3 ⋊ #4147 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C24 |
| Fitting | C3×C5×Q8 | C22 |
| hypercenter | Q8 | D3×D5 |
| hypocenter | C3×C5 | 2−1+4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
292 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 4 generators on 24 points:
(1,2,4,5,3)(6,7,8)(9,13,14,11)(10,15,17,21)(12,18,20,19)(16,23,24,22);(9,12,14,20)(10,16,17,24)(11,18,13,19)(15,22,21,23);(7,8)(9,11,14,13)(10,15,17,21)(12,19,20,18)(16,23,24,22);(2,3)(4,5)(9,10)(11,15)(12,16)(13,21)(14,17)(18,22)(19,23)(20,24)Polycyclic presentation:
2^5,3,5|b^2=e,c^2=e,d^2=e,d^a=de,g^a=g^4,c^b=ce,f^b=f^2,d^c=de57 conjugacy classes — view the full character table.
| maximal subgroup of | #11754082, #11754088, #11754092, … |
| maximal normal subgroup of | #11754082, #11754088, #11754092, … |
| maximal quotient of | #11753318, #11753319, #11753326, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 1104);