Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Hall | C3×C52 ⋊ C22⋊C4 |
| Sylow | C52 ⋊ (C2×D3)⋊C4 |
| Sylow | C3 ⋊ (C2×AGL(1,𝔽5))⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | ΓL(1,𝔽16)⋊D5 |
| commutator, Fitting, socle | C2×C3×C52 | C2×C4 |
| prime radical | C2×D5×D15 | C2 |
| cosocle | C2×C3×C5×D5 | C22 |
| hypocenter | C3×C52 | C22⋊C4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
146 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 146 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 120 |
Polycyclic presentation:
2^4,3,5^2|b^2=c,b^a=bd,e^a=e^2,g^a=g^4,f^b=f^2,g^b=g^4,f^c=f^460 conjugacy classes — view the full character table.
| maximal subgroup of | D5×(C2×D3)⋊AGL(1,𝔽5), D3×(C2×AGL(1,𝔽5))⋊D5, #72230105 |
| maximal normal subgroup of | D5×(C2×D3)⋊AGL(1,𝔽5), D3×(C2×AGL(1,𝔽5))⋊D5, #72230105 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 589);