Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3×C9 · Schur multiplier (H2) = C22 |
| direct product | C2×Q8×C27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C9 | C22×C32 |
| commutator | C2×C3 | C23×C3×C9 |
| Frattini, cosocle | C2×C9 | C23×C32 |
| socle | C22×C3 | C22×C3×C9 |
| prime radical | C22×C32 | C22×C9 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C9 ◁ G | |
228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 37 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 37 points:
(1,2)(11,12,15,13,17,23,19,26,31,14,18,22,20,27,30,28,34,36,21,16,24,29,25,32,35,33,37);(3,4,5,6)(7,10,9,8)(12,16,18)(15,22,24)(17,25,27)(23,30,32)(26,33,34)(31,36,37);(3,7,5,9)(4,8,6,10)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1223);