SL(2,ℤ/4) #228

order	48
isoclinism	isoclinic class of S₄
automorphism	Aut(G) = C₂×S₄ (order 48) · Inn(G) = S₄ · Out(G) = C₂
holomorph	G ⋊ Aut(G) = #72215293
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₂

Aliases

special linear	SL(2,ℤ/4)
symplectic	Sp(2,ℤ/4)
unambiguous semidirect product	A₄⋊C₄

A₄ ⋊ C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	S₄
commutator, hypocenter	A₄	C₄
Fitting, socle	C₂³	D₃
prime radical, cosocle	C₂×A₄	C₂
LCS	G ▷ A₄
UCS	C₁ ◁ C₂

19 conjugacy classes of subgroups.

minimal degree	μ(G) = 8
transitive degree	μ_t(G) = 12

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

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

Polycyclic presentation:

2^2,3,2^2|a^2=b,c^a=c^2,d^a=e,e^a=d,d^c=e,e^c=de

Here i is the imaginary unit (i² = −1).

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(48, 30);