Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #634⋊C3 |
| automorphism | Aut(G) = D4×((C3⋊D3)⋊C3)⋊C2 (order 864) · Inn(G) = C2×(C3⋊D3)⋊C3 · Out(G) = C23 |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #634⋊C3 |
| Sylow | H(3,𝔽3) ⋊ Q8 |
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:
(1,3,4,8)(2,5,6,7)(9,10,16)(11,13,15)(12,14,17);(1,2,4,6)(3,7,8,5)(10,12)(11,14)(13,15)(16,17)Polycyclic presentation:
2^2,3,2,3^2|a^2=d,b^2=d,b^a=bd,e^a=e^2,f^a=f^2,e^c=ef31 conjugacy classes — view the full character table.
| maximal subgroup of | #88431, #88433, #88437, … |
| maximal normal subgroup of | #88431, #88433, #88437, … |
| maximal quotient of | #3750⋊C3, C4⋊((C3⋊Dic3)⋊C3), C2×#634⋊C3, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 49);