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×C3 |
| direct product | C2×Q8×C9⋊C9 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C22×C32 | C22×C32 |
| commutator | C2×C3 | C23×C3×C9 |
| Frattini, cosocle | C2×C32 | C23×C32 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
323 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 1296 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(12,14,17)(15,19,18)(20,21,23,22,24,26,25,27,28);(3,4,5,6)(7,10,9,8)(11,12,15,13,14,18,16,17,19);(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, 1221);