Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C33 · Schur multiplier (H2) = C22×C34 |
| direct product | C2×C3×Q8×H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C32 | C22×C32 |
| commutator, Frattini, cosocle | C2×C3 | C23×C33 |
| prime radical | C22×C3×H(3,𝔽3) | C22 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C22×C32 ◁ G | |
1064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 22 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 22 points:
(1,2)(11,12,13);(3,4,5,6)(7,10,9,8)(14,15,17)(16,19,21)(18,22,20);(3,7,5,9)(4,8,6,10)(15,18,19)(17,21,22)330 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 2698);