C₄⋊SU(3,𝔽₂) #11722319

order	864
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

Fitting	C₄×H(3,𝔽₃) ⋊ Q₈
Sylow	H(3,𝔽₃) ⋊ C₄⋊Q₈

	subgroup	G/H
center, Frattini, socle	C₂×C₃	C₂×PSU(3,𝔽₂)
commutator, prime radical, cosocle	C₂×#272	C₂³
Fitting	C₄×H(3,𝔽₃)	Q₈
hypercenter	C₃×C₄	PSU(3,𝔽₂)
hypocenter	H(3,𝔽₃)	C₄⋊Q₈
LCS	G ▷ C₂×#272 ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄

118 conjugacy classes of subgroups.

μ_t(G) = 108

Polycyclic presentation:

2^5,3^3|a^2=d,b^2=d,c^2=e,b^a=bd,f^a=f^2g^2h,g^a=f^2g,c^b=ce,f^b=fg^2h^2,g^b=f^2g^2,f^d=f^2h,g^d=g^2h^2,g^f=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(864, 2286);