D₇₇ #3900

order	154
isoclinism	isoclinic class of D₇₇
automorphism	Aut(G) = AGL(1,𝔽₇)×AGL(1,𝔽₁₁) (order 4620) · Inn(G) = G · Out(G) = C₂×C₃×C₅
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₇₇
unambiguous semidirect product	C₇⋊D₁₁
unambiguous semidirect product	C₁₁⋊D₇

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₁	C₂
LCS	G ▷ C₇×C₁₁
UCS	C₁

8 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 77

A minimal faithful permutation representation — 2 generators on 18 points:

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

Polycyclic presentation:

2,7,11|b^a=b^6,c^a=c^10

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(154, 3);