Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | Aut(G) order 269568 |
| homology | Abelianization (H1) = C22×C33 · Schur multiplier (H2) = C33 |
| direct product | C33×Q8 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C2×C33 | C22 |
| commutator, Frattini, cosocle | C2 | C22×C33 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C33 ◁ G | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 3 generators on 17 points:
(1,2,3)(10,11,12,13)(14,17,16,15);(4,5,6)(10,14,12,16)(11,15,13,17);(7,8,9)135 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 152);