Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| isoclinism | isoclinic class of (C5⋊S4)⋊D5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | (C5⋊S4)⋊D5 |
| Fitting, socle | C22×C52 ⋊ C2×D3 |
| derived | C52×A4 ⋊ C22 |
| Sylow | C52 ⋊ C2×S4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C52×A4 | C22 |
| Fitting, socle | C22×C52 | C2×D3 |
| LCS | G ▷ C52×A4 | |
| UCS | C1 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 40 |
Polycyclic presentation:
2^2,3,5^2,2^2|c^a=c^2,d^a=d^4,f^a=fg,d^b=d^4,e^b=e^4,f^c=g,g^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #72213721, D52×S4, #72213957, … |
| maximal normal subgroup of | D52×S4, #72213957, #72214019, … |
| maximal quotient of | #72213721, C2×(C5⋊S4)⋊D5, C3×(C5⋊S4)⋊D5, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 984);