Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Hall | C3×C52 ⋊ D8 |
| Sylow | C52 ⋊ #213 |
| Sylow | C3 ⋊ #87771 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10597127 |
| commutator, Fitting, cosocle | C3×C4×C52 | C22 |
| Frattini, hypercenter | C4 | D15⋊D5 |
| socle | C2×C3×C52 | D4 |
| hypocenter | C3×C52 | D8 |
| LCS | G ▷ C3×C4×C52 ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
120 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 31 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|c^2=d,b^a=bc,c^a=cd,e^a=e^2,f^a=f^4,c^b=cd,f^b=f^4,g^b=g^4102 conjugacy classes — view the full character table.
| maximal subgroup of | #72230821, D3×#87771, #72230841, … |
| maximal normal subgroup of | #72230821, D3×#87771, #72230841, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 363);