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 ⋊ #87831 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | #10597182 |
| commutator, Fitting, socle | C2×C3×C52 | C2×C4 |
| prime radical | C2×D15⋊D5 | C2 |
| cosocle | C2×C3×C5⋊D5 | C22 |
| hypocenter | C3×C52 | C22⋊C4 |
| LCS | G ▷ C2×C3×C52 ▷ C3×C52 | |
| UCS | C1 ◁ C2 | |
168 conjugacy classes of subgroups; only the 31 normal subgroups are shown.
This group has 168 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) = 60 |
Polycyclic presentation:
2^4,3,5^2|b^2=c,b^a=bd,e^a=e^2,f^a=f^4,f^b=f^2,g^b=g^2,f^c=f^4,g^c=g^442 conjugacy classes — view the full character table.
| maximal subgroup of | #72224577, D3×#87831, #72224585, … |
| maximal normal subgroup of | #72224577, D3×#87831, #72224585, … |
| maximal quotient of | C3×#11925355, #72229372, C2×#11925355, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 565);