Finite Group DatabasebetaFinite Group Databasebeta| order | 189 |
| isoclinism | isoclinic class of #4427 |
| automorphism | Aut(G) = C3×D3×AGL(1,𝔽7) (order 756) · Inn(G) = C3×ΓL(1,𝔽8) · Out(G) = C22×C3 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Fitting | C7×C9 ⋊ C3 |
| Sylow | C7 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C32 |
| Fitting | C7×C9 | C3 |
| prime radical | C3×ΓL(1,𝔽8) | C3 |
| hypercenter | C9 | ΓL(1,𝔽8) |
| hypocenter | C7 | 3−1+2 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C9 | |
16 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 63 |
A minimal faithful permutation representation — 2 generators on 16 points:
(1,2,3,5,4,6,7)(8,9,12,13,14,16,10,11,15);(2,3,4)(5,7,6)(9,11,14)(12,16,15)Polycyclic presentation:
3^3,7|b^3=c^2,b^a=bc,d^a=d^229 conjugacy classes — view the full character table.
| maximal subgroup of | #67355, ΓL(1,𝔽64), #67365, … |
| maximal normal subgroup of | #67355, ΓL(1,𝔽64), #67365, … |
| maximal quotient of | C2×#4428, (C7⋊C9)⋊C9, #10588036, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(189, 5);