Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C52 · Schur multiplier (H2) = C5 |
| direct product | C3×C52×Q16 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C52 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C52 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C52 ◁ C3×C4×C52 ◁ G | |
144 conjugacy classes of subgroups.
| minimal degree | μ(G) = 29 |
| transitive degree | μt(G) = 1200 |
A minimal faithful permutation representation — 2 generators on 29 points:
(1,2,3)(4,5,6,7,8)(14,15,16,17,18,19,20,21)(22,29,28,27,26,25,24,23);(9,10,11,12,13)(14,22,18,26)(15,23,19,27)(16,24,20,28)(17,25,21,29)525 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 472);