order	208
nilpotency class	3
isoclinism	isoclinic class of D₈
automorphism	Aut(G) = C₃×C₄×AGL(1,ℤ/8) (order 384) · Inn(G) = D₄ · Out(G) = C₂²×C₃×C₄
homology	Abelianization (H₁) = C₂²×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

C₁₃×Q₁₆

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂×C₁₃	D₄
commutator, Frattini, cosocle	C₄	C₂²×C₁₃
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₄×C₁₃ ◁ G

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 29
transitive degree	μ_t(G) = 208

A minimal faithful permutation representation — 2 generators on 29 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16,17,18,19,20,21)(22,29,28,27,26,25,24,23);(14,22,18,26)(15,23,19,27)(16,24,20,28)(17,25,21,29)

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(208, 27);