order	136
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = D₄×C₁₆ (order 128) · Inn(G) = C₂² · Out(G) = C₂×C₁₆
holomorph	G ⋊ Aut(G) = #453×AGL(1,𝔽₁₇)
homology	Abelianization (H₁) = C₂²×C₁₇ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₁₇
unambiguous semidirect product	C₄⋊(C₂×C₁₇)
unambiguous semidirect product	C₂²⋊(C₂×C₁₇)

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₇	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₁₇
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₇ ◁ G

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 21
transitive degree	μ_t(G) = 68

A minimal faithful permutation representation — 2 generators on 21 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(136, 10);