Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4×C25 · Schur multiplier (H2) = C22 |
| direct product | C3×C22⋊C4×C25 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C3×C25 | C22 |
| commutator | C2 | C2×C3×C4×C25 |
| Frattini, cosocle | C22×C5 | C22×C3×C5 |
| socle | C22×C3×C5 | C22×C5 |
| prime radical | C23×C3×C5 | C2×C5 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C3×C25 ◁ G | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 600 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3)(4,5,7,9)(6,8,10,11)(12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(5,8)(9,11)750 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 82);