C₃⋊#261 #3913

order	156
isoclinism	isoclinic class of C₃⋊#261
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = G · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₃	C₄
prime radical, cosocle	C₃×D₁₃	C₂
LCS	G ▷ C₃×C₁₃
UCS	C₁

12 conjugacy classes of subgroups.

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 39

A minimal faithful permutation representation — 2 generators on 16 points:

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

Polycyclic presentation:

2^2,3,13|a^2=b,c^a=c^2,d^a=d^5,d^b=d^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 10);