Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4184 |
| automorphism | Aut(G) = (#4176⋊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 | #731 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #731 | C2 |
| Frattini | C32 | C3⋊D3 |
| socle | C3 | #272 |
| LCS | G ▷ #731 | |
| UCS | C1 | |
28 conjugacy classes of subgroups.
| minimal degree | μ(G) = 27 |
| transitive degree | μt(G) = 27 |
A minimal faithful permutation representation — 3 generators on 27 points:
(2,6)(3,9)(4,5)(7,17)(8,19)(10,22)(11,12)(13,21)(14,16)(15,25)(18,27)(20,23)(24,26);(1,13,9,5,12,11,4,3,21)(2,10,25,8,24,18,14,7,20)(6,23,17,16,27,26,19,15,22);(1,2,6)(3,10,23)(4,14,19)(5,8,16)(7,15,12)(9,20,22)(11,25,17)(13,24,27)(18,26,21)Polycyclic presentation:
2,3^4|c^3=e,b^a=b^2,c^a=c^2e^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 | #4176⋊C2, C2×#4184, C3×#4184, … |
| maximal normal subgroup of | #4176⋊C2, C2×#4184, C3×#4184, … |
| maximal quotient of | #66331, C2×#4184, #92199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 20);