Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6229 |
| automorphism | Aut(G) = D4×AGL(2,𝔽3) (order 3456) · Inn(G) = C2×C3⋊D3 · Out(G) = C2×GL(2,𝔽3) |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ Q8 |
66 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 72 |
A minimal faithful permutation representation — 3 generators on 17 points:
(2,5,6)(4,8,9)(10,12,13,17)(11,14,15,16);(2,4)(5,9)(6,8)(10,11,13,15)(12,16,17,14);(1,2,4)(3,6,8)(5,9,7)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^2,e^d=ef31 conjugacy classes — view the full character table.
| maximal subgroup of | #88438, #88439, #88528, … |
| maximal normal subgroup of | #88438, #88439, #88528, … |
| maximal quotient of | #88540, C4⋊#1084, C2×#6229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 66);