SL(2,𝔽₃)⋊C₈ #4624

order	192
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂³×S₄ (order 192) · Inn(G) = S₄ · Out(G) = C₂³
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₄	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₈
Fitting	C₄×Q₈	D₃
socle	C₂²	SL(2,ℤ/4)
prime radical	C₂×SL(2,𝔽₃)	C₄
cosocle	C₄×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₄

42 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^3,3,2^3|a^2=b,b^2=c,e^2=g,f^2=g,d^a=d^2,e^a=f,f^a=e,e^d=fg,f^d=ef,f^e=fg

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 183);