Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4207 |
| automorphism | Aut(G) = (#4207⋊C2)⋊C3 (order 972) · Inn(G) = G · Out(G) = C2×C3 |
| homology | Abelianization (H1) = C2×C3 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, Sylow | #737 ⋊ C2 |
| derived | C3×C9 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C9 | C2×C3 |
| Frattini, socle | C3 | C3×C3⋊D3 |
| Fitting | #737 | C2 |
| LCS | G ▷ C3×C9 | |
| UCS | C1 | |
39 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,5)(3,19,21,9,10,20)(4,11)(6,8)(7,24,26,15,16,25)(12,27,14,17,22,23)(13,18);(1,3,9)(2,7,17)(4,10,20)(5,12,15)(6,14,24)(8,16,23)(11,21,19)(13,22,25)(18,26,27);(1,18,5,11,8,6,4,2,13)(3,26,12,21,16,14,10,7,22)(9,27,15,19,23,24,20,17,25)Polycyclic presentation:
2,3^4|c^3=e^2,c^a=c^2e,d^a=d^2,e^a=e^2,d^b=de21 conjugacy classes — view the full character table.
| maximal subgroup of | #4207⋊C2, C2×#4207, #92182, … |
| maximal normal subgroup of | #4207⋊C2, C2×#4207, C3×#4207, … |
| maximal quotient of | #66409, C2×#4207, #92294, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 43);