C₂²⋊(C₁₃⋊C₃) #3917

order	156
isoclinism	isoclinic class of C₂²⋊(C₁₃⋊C₃)
automorphism	Aut(G) = A₄×AGL(1,𝔽₁₃) (order 1872) · Inn(G) = G · Out(G) = C₃×C₄
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

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

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

10 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 14);