#2202⋊C₃ #68041

order	384
automorphism	Aut(G) = #73026 (order 384) · Inn(G) = B(4,𝔽₂)⋊C₃ · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

Fitting, Sylow	#2202 ⋊ C₃
derived	H(3,𝔽₄) ⋊ C₂×C₃

	subgroup	G/H
center, socle	C₂	B(4,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	H(3,𝔽₄)	C₂×C₃
Frattini	C₂×Q₈	C₂×A₄
Fitting	#2202	C₃
hypercenter	C₂²	#120⋊C₃
LCS	G ▷ H(3,𝔽₄)
UCS	C₁ ◁ C₂ ◁ C₂²

78 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 2 generators on 16 points:

(1,2)(3,8)(4,12,11,6,5,14)(7,13,16,10,15,9);(2,15)(5,10)(6,12)(7,16)(8,14)(11,13)

Polycyclic presentation:

2,3,2^6|e^2=h,f^2=h,c^a=ceg,d^a=dfh,f^a=fh,g^a=gh,c^b=dh,d^b=cdh,e^b=fgh,f^b=ef,e^c=eh,f^c=fgh,e^d=eg,f^d=fh,f^e=fh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 617);