Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4183 |
| automorphism | Aut(G) = (#4174⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, derived, Sylow | AΣL(1,𝔽27) ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | AΣL(1,𝔽27) | C2 |
| Frattini | C32 | C3⋊D3 |
| socle | C3 | #272 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 | |
40 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 3 generators on 9 points:
(2,5)(3,4)(6,9)(7,8);(2,6,7)(5,8,9);(1,2,5)(3,6,8)(4,7,9)Polycyclic presentation:
2,3^4|b^a=b^2,c^a=c^2,d^a=de^2,e^a=e^2,c^b=cd,d^b=de13 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C2, C2×#4183, C3×#4183, … |
| maximal normal subgroup of | #4174⋊C2, C2×#4183, C3×#4183, … |
| maximal quotient of | #66330, C2×#4183, #92198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 19);