Finite Group DatabasebetaFinite Group Databasebeta| order | 162 |
| isoclinism | isoclinic class of #4185 |
| automorphism | Aut(G) = #72243845 (order 2916) · Inn(G) = G · Out(G) = C3×D3 |
| homology | Abelianization (H1) = C2 · Schur multiplier (H2) = C3 |
No aliases for this group.
| Fitting, derived, Sylow | #732 ⋊ C2 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, Fitting, cosocle, hypocenter | #732 | C2 |
| Frattini | C32 | C3⋊D3 |
| socle | C3 | #272 |
| LCS | G ▷ #732 | |
| UCS | C1 | |
32 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)(3,9)(4,11)(6,14)(7,13)(8,18)(10,21)(12,16)(15,20)(17,26)(19,25)(22,23)(24,27);(1,3,9)(2,6,16)(4,10,22)(5,12,14)(7,19,27)(8,17,20)(11,23,21)(13,24,25)(15,26,18);(1,2,5)(3,6,15)(4,8,13)(7,18,11)(9,20,14)(10,17,12)(16,26,21)(19,24,23)(22,27,25)Polycyclic presentation:
2,3^4|b^a=b^2,c^a=c^2,e^a=e^2,c^b=cd,d^b=de,d^c=de^213 conjugacy classes — view the full character table.
| maximal subgroup of | #4178⋊C2, C2×#4185, #92191, … |
| maximal normal subgroup of | #4178⋊C2, C2×#4185, #92191, … |
| maximal quotient of | #66332, C2×#4185, #92198, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(162, 21);