Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×H(3,𝔽3) |
| automorphism | Aut(G) = S4×#272 (order 1296) · Inn(G) = C22×C32 · Out(G) = D32 |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C1 |
| direct product | Q8×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2×C3 | C22×C32 |
| prime radical | C2×C32 | C22×C3 |
| LCS | G ▷ C2×C3 ▷ C1 | |
| UCS | C1 ◁ C2×C3 ◁ G | |
48 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 2 generators on 17 points:
(1,2,4,3,6,8,7,5,9)(10,11,13,15)(12,16,17,14);(2,5,6)(4,8,9)(10,12,13,17)(11,14,15,16)55 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 81);