C₄⋊PSU(3,𝔽₂) #64245

order	288
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₃

Aliases

Fitting	C₃²×C₄ ⋊ Q₈
Sylow	C₃² ⋊ C₄⋊Q₈

	subgroup	G/H
center, Frattini	C₂	C₂×PSU(3,𝔽₂)
commutator, prime radical, cosocle	C₂×C₃⋊D₃	C₂³
Fitting	C₃²×C₄	Q₈
socle	C₂×C₃²	C₂×Q₈
hypercenter	C₄	PSU(3,𝔽₂)
hypocenter	C₃²	C₄⋊Q₈
LCS	G ▷ C₂×C₃⋊D₃ ▷ C₃²
UCS	C₁ ◁ C₂ ◁ C₄

76 conjugacy classes of subgroups.

minimal degree	μ(G) = 13
transitive degree	μ_t(G) = 36

A minimal faithful permutation representation — 3 generators on 13 points:

(1,4,9)(2,7,3)(5,8,6)(10,11,12,13);(2,3,5,8)(4,6,9,7)(11,13);(2,4,5,9)(3,7,8,6)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(288, 893);