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) = #72253878 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #732 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×D3 |
| Fitting | #732 | C2 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
23 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,9,15,5,8,6,4,2,16)(3,26,18,21,7,27,13,20,14,10,19,25,12,17,22,23,11,24);(1,3,10)(2,7,17)(4,13,23)(5,12,21)(6,14,24)(8,19,26)(9,11,20)(15,22,27)(16,18,25)Polycyclic presentation:
2,3^4|b^3=e,c^a=c^2,d^a=d^2e,c^b=cd,d^c=de22 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊C2, C2×#4178, #92216, … |
| maximal normal subgroup of | #4178⋊C2, C2×#4178, C3×#4178, … |
| maximal quotient of | #66325, C2×#4178, #92160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 14);