#395⋊C₃ #4637

order	192
automorphism	Aut(G) = C₂×#4625⋊C₂ (order 768) · Inn(G) = #120⋊C₃ · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

#395 ⋊ C₃

	subgroup	G/H
center, hypercenter	C₂	#120⋊C₃
commutator, cosocle, hypocenter	C₂²×Q₈	C₂×C₃
Frattini, socle	C₂³	C₂×A₄
Fitting	#395	C₃
prime radical	C₂²⋊SL(2,𝔽₃)	C₂
LCS	G ▷ C₂²×Q₈
UCS	C₁ ◁ C₂

47 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 24

A minimal faithful permutation representation — 2 generators on 24 points:

(1,6,19,8,3,15,5,17,9,2,12,22)(4,10,24,7,11,18,14,16,21,13,20,23);(1,13,5,7)(2,4,8,14)(3,6,12,17)(9,21,19,24)(10,20,16,11)(15,23,22,18)

Polycyclic presentation:

2,3,2^5|a^2=g,c^2=g,d^2=g,c^a=ce,d^a=df,c^b=d,d^b=cd,e^b=f,f^b=ef,d^c=dg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 196);