Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) = C2×S4×#272 (order 2592) · Inn(G) = C22 · Out(G) = C2×D3×#272 |
| homology | Abelianization (H1) = C22×C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×Q8×C9 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3×C9 | C22 |
| commutator | C2 | C22×C3×C9 |
| Frattini, cosocle | C2×C3 | C22×C32 |
| socle, prime radical | C2×C32 | C22×C3 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C3×C9 ◁ G | |
60 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 2 generators on 20 points:
(1,2,3)(13,14,15,16)(17,20,19,18);(4,5,6,7,8,9,10,11,12)(13,17,15,19)(14,18,16,20)135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 79);