Finite Group DatabasebetaFinite Group Databasebeta| order | 816 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C17 · Schur multiplier (H2) = C1 |
| direct product | C3×Q16×C17 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C3×C17 | D4 |
| commutator, Frattini, cosocle | C4 | C22×C3×C17 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C17 ◁ C3×C4×C17 ◁ G | |
36 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 816 |
A minimal faithful permutation representation — 2 generators on 36 points:
(1,2,3)(4,5,6,7,8,9,10,11)(12,19,18,17,16,15,14,13)(20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36);(4,12,8,16)(5,13,9,17)(6,14,10,18)(7,15,11,19)357 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(816, 88);