Finite Group DatabasebetaFinite Group Databasebeta| order | 480 |
| isoclinism | isoclinic class of (C2×D3)⋊AGL(1,𝔽5) |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×AGL(1,𝔽5))⋊Dic3 |
| Hall | C3×C5 ⋊ #95 |
| Sylow | C5 ⋊ C3⋊#95 |
| Sylow | C3 ⋊ (C2×AGL(1,𝔽5))⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×AGL(1,𝔽5) |
| commutator | C2×C3×C5 | C42 |
| Fitting, socle | C22×C3×C5 | C2×C4 |
| prime radical, cosocle | C22×C3×D5 | C22 |
| hypocenter | C3×C5 | #95 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 | |
152 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 152 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) = 16 |
| transitive degree | μt(G) = 120 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,5,4)(7,8)(9,10,11,12)(14,16);(2,3,4,5)(6,7,8)(13,14)(15,16)Polycyclic presentation:
2^5,3,5|a^2=c,b^2=d,b^a=be,g^a=g^2,f^b=f^2,g^c=g^448 conjugacy classes — view the full character table.
| maximal subgroup of | #11751745, #11751746, #11751747, … |
| maximal normal subgroup of | #11751745, #11751746, #11751747, … |
| maximal quotient of | (C2×AGL(1,𝔽5))⋊(C3⋊C8), #11746517, #11746518, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(480, 243);