C₂×C₃×D₁₃ #3918

order	156
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₃) (order 624) · Inn(G) = D₁₃ · Out(G) = C₂³×C₃
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂

Aliases

C₂×C₃×D₁₃

	subgroup	G/H
center, hypercenter	C₂×C₃	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂²×C₃
Frattini	C₁	G
Fitting, socle	C₂×C₃×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₂×C₃

20 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 78

A minimal faithful permutation representation — 2 generators on 18 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(156, 15);