Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4174 |
| automorphism | Aut(G) = C3×((C3⋊D3)⋊C3)⋊C2 (order 324) · Inn(G) = (C3⋊D3)⋊C3 · Out(G) = C2×C3 |
| holomorph | G ⋊ Aut(G) = #72253868 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | AΣL(1,𝔽27) ⋊ C2 |
| derived | H(3,𝔽3) ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×D3 |
| Fitting | AΣL(1,𝔽27) | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
34 conjugacy classes of subgroups.
| minimal degree | μ(G) = 9 |
| transitive degree | μt(G) = 9 |
A minimal faithful permutation representation — 2 generators on 9 points:
(2,8,7,5,6,9);(1,2,5)(3,7,9)(4,6,8)Polycyclic presentation:
2,3^4|c^a=c^2,d^a=d^2e,c^b=cd,d^c=de22 conjugacy classes — view the full character table.
| maximal subgroup of | #4174⋊C2, C2×#4174, C3×#4174, … |
| maximal normal subgroup of | #4174⋊C2, C2×#4174, C3×#4174, … |
| maximal quotient of | #66321, C2×#4174, #92159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 10);