D₆₉ #3640

order	138
isoclinism	isoclinic class of D₆₉
automorphism	Aut(G) = D₃×AGL(1,𝔽₂₃) (order 3036) · Inn(G) = G · Out(G) = C₂×C₁₁
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₉
unambiguous semidirect product	C₃⋊D₂₃
unambiguous semidirect product	C₂₃⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₂₃	C₂
LCS	G ▷ C₃×C₂₃
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 69

A minimal faithful permutation representation — 2 generators on 26 points:

(1,2,3)(4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26);(2,3)(5,26)(6,25)(7,24)(8,23)(9,22)(10,21)(11,20)(12,19)(13,18)(14,17)(15,16)

Polycyclic presentation:

2,3,23|b^a=b^2,c^a=c^22

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(138, 3);