#2205⋊C₃ #68042

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	#2205 ⋊ C₃
derived	Q₈² ⋊ C₂×C₃

	subgroup	G/H
center, socle	C₂	B(4,𝔽₂)⋊C₃
commutator, cosocle, hypocenter	Q₈²	C₂×C₃
Frattini	C₂×Q₈	C₂×A₄
Fitting	#2205	C₃
hypercenter	C₂²	#120⋊C₃
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂ ◁ C₂²

69 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)(3,13,5,8,4,7)(6,10)(9,11,15,14,16,12);(1,11,6,5)(2,9,10,13)(3,14,12,4)(7,15,16,8)

Polycyclic presentation:

2,3,2^6|c^2=h,d^2=h,e^2=h,f^2=h,c^a=ceh,d^a=dfgh,e^a=eh,g^a=gh,c^b=d,d^b=cdh,e^b=fg,f^b=efgh,d^c=dh,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, 618);