Finite Group DatabasebetaFinite Group Databasebeta| order | 189 |
| isoclinism | isoclinic class of #4427 |
| automorphism | Aut(G) = H(3,𝔽3)×AGL(1,𝔽7) (order 1134) · Inn(G) = C3×ΓL(1,𝔽8) · Out(G) = C2×C32 |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C1 |
No aliases for this group.
| Sylow | C7 ⋊ 3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×ΓL(1,𝔽8) |
| commutator, socle, cosocle | C3×C7 | C32 |
| Fitting, prime radical | C32×C7 | C3 |
| hypercenter | C32 | ΓL(1,𝔽8) |
| hypocenter | C7 | 3−1+2 |
| LCS | G ▷ C3×C7 ▷ C7 | |
| UCS | C1 ◁ C3 ◁ C32 | |
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)(9,12,13)(11,15,16);(2,3,4)(5,7,6)(8,9,11,10,13,15,14,12,16)Polycyclic presentation:
3^3,7|a^3=c,b^a=bc,d^a=d^229 conjugacy classes — view the full character table.
| maximal subgroup of | #4430⋊C2, C2×#4430, #10588037, … |
| maximal normal subgroup of | #4430⋊C2, C2×#4430, #10588037, … |
| maximal quotient of | C2×#4430, (C7⋊C9)⋊C9, C32⋊(C7⋊C9), … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(189, 7);