Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 1 |
| isoclinism | isoclinic class of C1 |
| automorphism | Aut(G) order 62208 |
| homology | Abelianization (H1) = G · Schur multiplier (H2) = C23×C3 |
| abelian | C22×C3×C4×C27 |
| multiplicative | (ℤ/2289)ˣ, (ℤ/2835)ˣ, (ℤ/3052)ˣ, (ℤ/3924)ˣ, (ℤ/4212)ˣ, (ℤ/4578)ˣ, (ℤ/5670)ˣ |
| subgroup | G/H | |
|---|---|---|
| center | G | C1 |
| commutator | C1 | G |
| Frattini, cosocle | C2×C9 | C23×C32 |
| socle, prime radical | C23×C32 | C2×C9 |
| LCS | G ▷ C1 | |
| UCS | C1 ◁ G | |
378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 1296 |
A minimal faithful permutation representation — 3 generators on 38 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);(5,6)(36,37,38);(7,8)1296 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1204);