Finite Group DatabasebetaFinite Group Databasebeta| order | 189 |
| isoclinism | isoclinic class of ΓL(1,𝔽8) |
| automorphism | Aut(G) = AGL(1,𝔽7)×#272 (order 2268) · Inn(G) = ΓL(1,𝔽8) · Out(G) = C2×#272 |
| homology | Abelianization (H1) = C3×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×C7⋊C9 |
| unambiguous semidirect product | (C3×C7)⋊C9 |
| derived, Sylow | C7 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32 | ΓL(1,𝔽8) |
| commutator, hypocenter | C7 | C3×C9 |
| Frattini | C3 | C3×ΓL(1,𝔽8) |
| Fitting, socle, prime radical | C32×C7 | C3 |
| cosocle | C3×C7 | C32 |
| LCS | G ▷ C7 | |
| UCS | C1 ◁ C32 | |
20 conjugacy classes of subgroups.
| minimal degree | μ(G) = 19 |
| transitive degree | μt(G) = 189 |
A minimal faithful permutation representation — 2 generators on 19 points:
(2,3,4)(5,7,6)(11,12,14,13,15,17,16,18,19);(1,2,3,5,4,6,7)(8,9,10)45 conjugacy classes — view the full character table.
| maximal subgroup of | C3×D7⋊C9, D3×C7⋊C9, C3⋊(D7⋊C9), … |
| maximal normal subgroup of | C3×D7⋊C9, D3×C7⋊C9, C3⋊(D7⋊C9), … |
| maximal quotient of | C2×C3×C7⋊C9, C3×C7⋊C27, #10588032, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(189, 6);