AGL(1,𝔽₉)⋊D₃ #89093

order	432
isoclinism	isoclinic class of AGL(1,𝔽₉)⋊D₃
automorphism	Aut(G) = D₃×AΓL(1,𝔽₉) (order 864) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

C₃³ ⋊ SD₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₃×#157	C₂²
Fitting, socle, hypocenter	C₃³	SD₁₆
prime radical	#157⋊D₃	C₂
LCS	G ▷ C₃×#157 ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁

54 conjugacy classes of subgroups.

minimal degree	μ(G) = 12
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 12 points:

(1,2,3)(4,7,11,6,12,5,10,9);(2,3)(6,10)(7,11)(8,12)

Polycyclic presentation:

2^4,3^3|b^2=cd,c^2=d,b^a=bcd,c^a=cd,f^a=e^2f^2,g^a=g^2,e^b=ef,f^b=e,e^c=ef^2,f^c=e^2f^2,e^d=e^2,f^d=f^2

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(432, 740);