Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C3×C4×C52 ⋊ C22 |
| Hall | C3×C52 ⋊ C2×D4 |
| Sylow | C52 ⋊ C2×D12 |
| Sylow | C3 ⋊ #87877 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D15⋊D5 |
| commutator, socle, cosocle | C2×C3×C52 | C23 |
| Fitting | C3×C4×C52 | C22 |
| hypercenter | C4 | D15⋊D5 |
| hypocenter | C3×C52 | C2×D4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 ◁ C4 | |
248 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 248 conjugacy classes of subgroups — too many to draw. Only its 51 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,c^a=cd,e^a=e^2,f^a=f^4,f^b=f^4,g^b=g^4102 conjugacy classes — view the full character table.
| maximal subgroup of | #72229438, #72229609, #72229658, … |
| maximal normal subgroup of | #72229438, #72229609, #72229658, … |
| maximal quotient of | C3×#11925655, #72231336, #72259975 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 865);