Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23×C5 |
No aliases for this group.
| Fitting | C3×C4×C52 ⋊ C22 |
| Hall | C3×C52 ⋊ C2×D4 |
| Sylow | C52 ⋊ D3×D4 |
| Sylow | C3 ⋊ D4×C5⋊D5 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D3×C5⋊D5 |
| commutator, socle, cosocle | C2×C3×C52 | C23 |
| Fitting | C3×C4×C52 | C22 |
| hypercenter | C4 | D3×C5⋊D5 |
| hypocenter | C3×C52 | C2×D4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
432 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 432 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 300 |
Polycyclic presentation:
2^4,3,5^2|c^2=d,c^a=cd,f^a=f^4,g^a=g^4,c^b=cd,e^b=e^2123 conjugacy classes — view the full character table.
| maximal subgroup of | D3×D4×C5⋊D5, C2×#11925606, #72230586, … |
| maximal normal subgroup of | D3×D4×C5⋊D5, C2×#11925606, #72230787, … |
| maximal quotient of | C2×#11925606, #72230586, #72230998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 816);