SL(2,𝔽₃)⋊C₁₆ #68004

order	384
isoclinism	isoclinic class of #226
automorphism	Aut(G) = C₂²×C₄×S₄ (order 384) · Inn(G) = S₄ · Out(G) = C₂²×C₄
homology	Abelianization (H₁) = C₁₆ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂×C₈	S₄
commutator, hypocenter	SL(2,𝔽₃)	C₁₆
Fitting	C₈×Q₈	D₃
socle	C₂²	A₄⋊C₈
prime radical	C₂×SL(2,𝔽₃)	C₈
cosocle	C₈×SL(2,𝔽₃)	C₂
LCS	G ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₈

54 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^4,3,2^3|a^2=b,b^2=c,c^2=d,f^2=h,g^2=h,e^a=e^2,f^a=gh,g^a=fh,f^e=g,g^e=fgh,g^f=gh

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(384, 580);