Finite Group DatabasebetaFinite Group Databasebeta| order | 648 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C34 · Schur multiplier (H2) = C2×C36 |
| direct product | C34×D4 |
| unambiguous semidirect product | C4⋊(C2×C34) |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C34 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C34 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C34 ◁ G | |
1696 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 324 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2,3)(13,14,15,16);(4,5,6)(14,16);(7,8,9);(10,11,12)405 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(648, 700);