D₅₇ #1176

order	114
isoclinism	isoclinic class of D₅₇
automorphism	Aut(G) = D₃×AGL(1,𝔽₁₉) (order 2052) · Inn(G) = G · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₅₇
unambiguous semidirect product	C₃⋊D₁₉
unambiguous semidirect product	C₁₉⋊D₃

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₁₉	C₂
LCS	G ▷ C₃×C₁₉
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 22
transitive degree	μ_t(G) = 57

A minimal faithful permutation representation — 2 generators on 22 points:

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

Polycyclic presentation:

2,3,19|b^a=b^2,c^a=c^18

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(114, 5);