C₃²×D₁₃ #6622

order	234
isoclinism	isoclinic class of D₁₃
automorphism	Aut(G) = GL(2,𝔽₃)×AGL(1,𝔽₁₃) (order 7488) · Inn(G) = D₁₃ · Out(G) = C₂×C₃×GL(2,𝔽₃)
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃

Aliases

C₃²×D₁₃

Fitting, socle, Hall	C₃²×C₁₃ ⋊ C₂
derived, Sylow	C₁₃ ⋊ C₂×C₃²

	subgroup	G/H
center, hypercenter	C₃²	D₁₃
commutator, cosocle, hypocenter	C₁₃	C₂×C₃²
Frattini	C₁	G
Fitting, socle	C₃²×C₁₃	C₂
LCS	G ▷ C₁₃
UCS	C₁ ◁ C₃²

24 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 117

A minimal faithful permutation representation — 2 generators on 19 points:

(4,5,6)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19);(1,2,3)(7,8,10,12,14,16,18,19,17,15,13,11,9)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(234, 11);