Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6308 |
| automorphism | Aut(G) order 2426112 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C33 |
No aliases for this group.
| Sylow | C33 ⋊ Q8 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊D3) |
| commutator, socle, prime radical, cosocle | C2×C33 | C22 |
| Fitting | C33×C4 | C2 |
| hypercenter | C4 | C3⋊(C3⋊D3) |
| hypocenter | C33 | Q8 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C4 | |
168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 17 |
| transitive degree | μt(G) = 216 |
A minimal faithful permutation representation — 4 generators on 17 points:
(1,3,4,8)(2,5,6,7)(15,16,17);(1,2,4,6)(3,7,8,5)(10,11)(13,14)(16,17);(12,13,14);(9,10,11)Polycyclic presentation:
2^3,3^3|a^2=c,b^2=c,b^a=bc,d^a=d^2,e^a=e^2,f^a=f^257 conjugacy classes — view the full character table.
| maximal subgroup of | #88794, #88796, #88799, … |
| maximal normal subgroup of | #88794, #88796, #88799, … |
| maximal quotient of | #88857, C4⋊(C3⋊(C3⋊Dic3)), C2×#6308, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 145);