Q₈⋊(C₁₃⋊C₃) #64610

order	312
isoclinism	isoclinic class of Q₈⋊(C₁₃⋊C₃)
automorphism	Aut(G) = A₄×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = C₂²⋊(C₁₃⋊C₃) · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	Q₈⋊(C₁₃⋊C₃)
unambiguous semidirect product	C₁₃⋊SL(2,𝔽₃)

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₂²⋊(C₁₃⋊C₃)
commutator, Fitting, cosocle, hypocenter	Q₈×C₁₃	C₃
socle	C₂×C₁₃	A₄
LCS	G ▷ Q₈×C₁₃
UCS	C₁ ◁ C₂

14 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 2 generators on 21 points:

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

Polycyclic presentation:

3,2^2,13,2|b^2=e,c^2=e,b^a=c,c^a=bc,d^a=d^3,c^b=ce

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 26);