Finite Group DatabasebetaFinite Group Databasebeta| order | 864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C33×C4 · Schur multiplier (H2) = C33 |
| direct product | C33×#108 |
| subgroup | G/H | |
|---|---|---|
| center | C33×C4 | D4 |
| commutator | C4 | C2×C33×C4 |
| Frattini, cosocle | C2×C4 | C22×C33 |
| socle | C2×C33 | C4⋊C4 |
| prime radical | C22×C33 | Q8 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C33×C4 ◁ C2×C33×C4 ◁ G | |
420 conjugacy classes of subgroups.
| minimal degree | μ(G) = 25 |
| transitive degree | μt(G) = 432 |
A minimal faithful permutation representation — 3 generators on 25 points:
(1,2,3)(10,11,14,17,15,18,13,23)(12,19,21,24,22,25,20,16);(4,5,6)(10,12,13,20,15,22,14,21)(11,16,17,19,18,24,23,25);(7,8,9)378 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(864, 2188);