Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C2×D3)⋊#1043 |
| Hall | C3×C52 ⋊ C22⋊C4 |
| Sylow | C52 ⋊ (C2×D3)⋊C4 |
| Sylow | C3 ⋊ C22⋊#1043 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D3×#1043 |
| commutator, Fitting, socle | C2×C3×C52 | C2×C4 |
| prime radical | C2×D3×C5⋊D5 | C2 |
| cosocle | C2×C3×C5⋊D5 | C22 |
| hypocenter | C3×C52 | C22⋊C4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
272 conjugacy classes of subgroups; only the 52 normal subgroups are shown.
This group has 272 conjugacy classes of subgroups — too many to draw. Only its 52 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|a^2=c,b^a=bd,f^a=f^2,g^a=g^2,e^b=e^2,f^c=f^4,g^c=g^454 conjugacy classes — view the full character table.
| maximal subgroup of | #72229531, D12×#1043, C3×(C2×D3)⋊#1043, … |
| maximal normal subgroup of | #72229531, D12×#1043, C3×(C2×D3)⋊#1043, … |
| maximal quotient of | C3×(C2×D3)⋊#1043, #72229567, #72229571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 546);