D₁₃⋊A₄ #64635

order	312
isoclinism	isoclinic class of D₁₃⋊A₄
automorphism	Aut(G) = A₄×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₁₃⋊A₄
unambiguous semidirect product	(C₂²⋊(C₁₃⋊C₃))⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₁₃	C₂×C₃
LCS	G ▷ C₂²×C₁₃
UCS	C₁

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 17
transitive degree	μ_t(G) = 52

A minimal faithful permutation representation — 2 generators on 17 points:

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

Polycyclic presentation:

2,3,2^2,13|e^a=e^12,c^b=d,d^b=cd,e^b=e^3

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 51);