B(4,𝔽₂)⋊C₃ #4642

order	192
automorphism	Aut(G) = C₂×#1020⋊C₂ (order 384) · Inn(G) = #120⋊C₃ · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²

Aliases

Fitting, Sylow	B(4,𝔽₂) ⋊ C₃
derived	H(5,𝔽₂) ⋊ C₂×C₃

	subgroup	G/H
center, socle, hypercenter	C₂	#120⋊C₃
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂×C₃
Frattini	C₂³	C₂×A₄
Fitting	B(4,𝔽₂)	C₃
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂

58 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 8

A minimal faithful permutation representation — 2 generators on 8 points:

(2,7,6,5,3,8);(1,6)(2,3)(4,8)(5,7)

Polycyclic presentation:

2,3,2^5|c^a=ce,d^a=df,c^b=d,d^b=cd,e^b=f,f^b=efg,f^c=fg,e^d=eg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 201);