#458⋊C₃ #4643

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) = C₂

Aliases

#458 ⋊ C₃

	subgroup	G/H
center, socle, hypercenter	C₂	#120⋊C₃
commutator, cosocle, hypocenter	H(5,𝔽₂)	C₂×C₃
Frattini	C₂³	C₂×A₄
Fitting	#458	C₃
prime radical	#997	C₂
LCS	G ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂

45 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:

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

Polycyclic presentation:

2,3,2^5|a^2=g,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, 202);