Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C9×Q32 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | D8 |
| commutator | C8 | C22×C3×C9 |
| Frattini, cosocle | C3×C8 | C22×C32 |
| socle, prime radical | C2×C32 | C3×D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ C3×C4×C9 ◁ C3×C8×C9 ◁ G | |
120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 864 |
A minimal faithful permutation representation — 2 generators on 44 points:
(1,2,3)(13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28)(29,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30);(4,5,6,7,8,9,10,11,12)(13,29,21,37)(14,30,22,38)(15,31,23,39)(16,32,24,40)(17,33,25,41)(18,34,26,42)(19,35,27,43)(20,36,28,44)297 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 608);