Finite Group DatabasebetaFinite Group Databasebeta| order | 216 |
| isoclinism | isoclinic class of #6308 |
| automorphism | Aut(G) order 1213056 |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2×C33 |
No aliases for this group.
| Fitting | C22×C33 ⋊ C2 |
| Sylow | C33 ⋊ D4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×C3⋊(C3⋊D3) |
| commutator, socle, cosocle | C2×C33 | C22 |
| Fitting | C22×C33 | C2 |
| hypercenter | C22 | C3⋊(C3⋊D3) |
| hypocenter | C33 | D4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C22 | |
224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 13 |
| transitive degree | μt(G) = 108 |
A minimal faithful permutation representation — 4 generators on 13 points:
(2,4)(11,12,13);(1,2)(3,4)(6,7)(9,10)(12,13);(8,9,10);(5,6,7)Polycyclic presentation:
2^3,3^3|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 | #89028, #89031, D3×#638, … |
| maximal normal subgroup of | #89028, #89031, D3×#638, … |
| maximal quotient of | #88857, #88859, #88860, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(216, 149);