Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of ΓL(1,𝔽16)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
| unambiguous semidirect product | ΓL(1,𝔽16)⋊Dic5 |
| derived, Hall | C3×C52 ⋊ C42 |
| Sylow | C52 ⋊ C4×Dic3 |
| Sylow | C3 ⋊ Dic5×AGL(1,𝔽5) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽16)⋊D5 |
| commutator, hypocenter | C3×C52 | C42 |
| Fitting, socle | C2×C3×C52 | C2×C4 |
| prime radical, cosocle | C2×C3×C5×D5 | C22 |
| LCS | G ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
126 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 126 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 240 |
Polycyclic presentation:
2^4,3,5^2|a^2=d,b^2=c,e^a=e^2,g^a=g^4,f^b=f^2,g^b=g^4,f^c=f^466 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 588);