Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| automorphism | ? |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | C25⋊GL(2,𝔽3) |
| Fitting | Q8×C25 ⋊ D3 |
| derived | SL(2,𝔽3)×C25 ⋊ C2 |
| Sylow | C25 ⋊ GL(2,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C2 | A4⋊D25 |
| commutator, cosocle, hypocenter | SL(2,𝔽3)×C25 | C2 |
| Frattini, socle | C2×C5 | C5⋊S4 |
| Fitting | Q8×C25 | D3 |
| LCS | G ▷ SL(2,𝔽3)×C25 | |
| UCS | C1 ◁ C2 | |
48 conjugacy classes of subgroups.
| transitive degree | μt(G) = 200 |
Polycyclic presentation:
2,3,5^2,2^3|c^5=d,e^2=g,f^2=g,b^a=b^2,c^a=c^4d^4,d^a=d^4,e^a=ef,f^a=fg,e^b=f,f^b=efg,f^e=fg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 100);