D₉₁ #4396

order	182
isoclinism	isoclinic class of D₉₁
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,𝔽₁₃) (order 6552) · Inn(G) = G · Out(G) = C₃²×C₄
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₉₁
unambiguous semidirect product	C₇⋊D₁₃
unambiguous semidirect product	C₁₃⋊D₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃	C₂
LCS	G ▷ C₇×C₁₃
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 91

A minimal faithful permutation representation — 2 generators on 20 points:

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

Polycyclic presentation:

2,7,13|b^a=b^6,c^a=c^12

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(182, 3);