D₃×#261 #64630

order	312
isoclinism	isoclinic class of D₃×#261
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₃) (order 936) · Inn(G) = G · Out(G) = C₃
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₃×#261
unambiguous semidirect product	(C₃×#261)⋊C₂
unambiguous semidirect product	(C₃⋊#261)⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃×C₁₃	C₂×C₄
prime radical	D₃×D₁₃	C₂
cosocle	C₃×D₁₃	C₂²
LCS	G ▷ C₃×C₁₃
UCS	C₁

32 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)(5,8,9,13)(7,11,12,10)(14,15,16);(1,2,5,8,12,6,10,11,3,7,13,9,4)(15,16)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(312, 46);