Q₈⋊SL(2,𝔽₃) #5463

order	192
automorphism	Aut(G) = (#1020⋊C₃)⋊C₂ (order 576) · Inn(G) = C₂²⋊A₄ · Out(G) = C₂×D₃
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂²

Aliases

Q₈² ⋊ C₃

	subgroup	G/H
center, Frattini, socle, hypercenter	C₂²	C₂²⋊A₄
commutator, Fitting, cosocle, hypocenter	Q₈²	C₃
LCS	G ▷ Q₈²
UCS	C₁ ◁ C₂²

62 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

3,2^6|b^2=g,c^2=g,d^2=f,e^2=f,b^a=c,c^a=bc,d^a=e,e^a=de,c^b=cg,e^d=ef

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(192, 1022);