Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #272 |
| automorphism | Aut(G) = C22×AGL(2,𝔽3) (order 1728) · Inn(G) = C3⋊D3 · Out(G) = C22×S4 |
| homology | Abelianization (H1) = C8 · Schur multiplier (H2) = C1 |
No aliases for this group.
| derived, Sylow | H(3,𝔽3) ⋊ C8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C3×C4 | C3⋊D3 |
| commutator, hypocenter | H(3,𝔽3) | C8 |
| Fitting, cosocle | C4×H(3,𝔽3) | C2 |
| socle | C2×C3 | C3⋊Dic3 |
| prime radical | C2×H(3,𝔽3) | C4 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3×C4 | |
44 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,4)(5,9)(6,8)(10,11,12,14,13,15,16,17);(2,5,6)(4,8,9);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^3,3^3|a^2=b,b^2=c,d^a=d^2,e^a=e^2,e^d=ef40 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 17);