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) = #72253869 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting, Sylow | #731 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, socle, hypercenter | C3 | (C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | H(3,𝔽3) | C2×C3 |
| Frattini | C32 | C3×D3 |
| Fitting | #731 | C2 |
| prime radical | #272 | C3 |
| LCS | G ▷ H(3,𝔽3) | |
| UCS | C1 ◁ C3 | |
22 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 2 generators on 27 points:
(1,7,24,4,18,5,12,2,14)(3,15,13,20,17,27,10,19,16,8,6,23,11,26,21,22,9,25);(1,3,8)(2,6,15)(4,11,20)(5,13,23)(7,17,26)(9,19,18)(10,22,12)(14,21,27)(16,25,24)Polycyclic presentation:
2,3^4|b^3=e,c^a=c^2,d^a=d^2e^2,c^b=cd,d^c=de^222 conjugacy classes — view the full character table.
| maximal subgroup of | #4176⋊C2, C2×#4176, C3×#4176, … |
| maximal normal subgroup of | #4176⋊C2, C2×#4176, C3×#4176, … |
| maximal quotient of | #66323, C2×#4176, #92162, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 12);