C₁₃×#41 #6137

order	208
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×C₃×C₄×S₄ (order 576) · Inn(G) = C₂² · Out(G) = C₂×C₃×C₄×D₃
homology	Abelianization (H₁) = C₂³×C₁₃ · Schur multiplier (H₂) = C₂²

Aliases

	subgroup	G/H
center	C₄×C₁₃	C₂²
commutator, Frattini, cosocle	C₂	C₂³×C₁₃
socle	C₂×C₁₃	C₂³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ G

40 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 104

A minimal faithful permutation representation — 3 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 48);