Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C52 · Schur multiplier (H2) = C22×C5 |
| direct product | C3×C52×#41 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4×C52 | C22 |
| commutator, Frattini, cosocle | C2 | C23×C3×C52 |
| socle | C2×C3×C52 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C3×C4×C52 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 600 |
A minimal faithful permutation representation — 3 generators on 21 points:
(1,2,3)(4,5,6,7,8)(14,15)(16,19)(17,18)(20,21);(9,10,11,12,13)(15,18)(19,21);(14,16,17,20)(15,19,18,21)750 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 938);