D₁₁₉ #6637

order	238
isoclinism	isoclinic class of D₁₁₉
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,𝔽₁₇) (order 11424) · Inn(G) = G · Out(G) = C₃×C₁₆
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₁₉
unambiguous semidirect product	C₇⋊D₁₇
unambiguous semidirect product	C₁₇⋊D₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₇	C₂
LCS	G ▷ C₇×C₁₇
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 119

A minimal faithful permutation representation — 2 generators on 24 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);(2,7)(3,6)(4,5)(9,24)(10,23)(11,22)(12,21)(13,20)(14,19)(15,18)(16,17)

Polycyclic presentation:

2,7,17|b^a=b^6,c^a=c^16

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(238, 3);