C₂²×C₁₃⋊C₃ #3915

order	156
isoclinism	isoclinic class of C₁₃⋊C₃
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = C₁₃⋊C₃ · Out(G) = C₄×D₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₂²×C₁₃⋊C₃

Fitting, socle, Hall	C₂²×C₁₃ ⋊ C₃
derived, Sylow	C₁₃ ⋊ C₂²×C₃

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

20 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:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 12);