C₇⋊H(3,𝔽₃) #4431

order	189
isoclinism	isoclinic class of #4427
automorphism	Aut(G) = AGL(1,𝔽₇)×(C₃⋊D₃)⋊C₃ (order 2268) · Inn(G) = C₃×ΓL(1,𝔽₈) · Out(G) = C₂×C₃×D₃
homology	Abelianization (H₁) = C₃² · Schur multiplier (H₂) = C₃²

Aliases

unambiguous semidirect product	C₇⋊H(3,𝔽₃)
unambiguous semidirect product	C₃²⋊ΓL(1,𝔽₈)
unambiguous semidirect product	(C₃×ΓL(1,𝔽₈))⋊C₃

Fitting	C₃²×C₇ ⋊ C₃
Sylow	C₇ ⋊ H(3,𝔽₃)

22 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(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)(12,16,14)

Polycyclic presentation:

3^3,7|b^a=bc,d^a=d^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(189, 8);