Finite Group DatabasebetaFinite Group Databasebeta| order | 1200 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 7680 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23 |
| abelian | C22×C3×C4×C25 |
| multiplicative | (ℤ/2121)ˣ, (ℤ/2265)ˣ, (ℤ/2416)ˣ, (ℤ/2625)ˣ, (ℤ/2828)ˣ, (ℤ/3020)ˣ, (ℤ/3500)ˣ, (ℤ/3636)ˣ, (ℤ/4242)ˣ, (ℤ/4500)ˣ, (ℤ/4530)ˣ, (ℤ/5250)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C5 | C23×C3×C5 |
| socle, prime radical | C23×C3×C5 | C2×C5 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 1200 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(9,10,11)(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);(5,6);(7,8)1200 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1200, 179);