order	152
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	Aut(G) = C₂×D₄×C₉ (order 144) · Inn(G) = C₂² · Out(G) = C₂²×C₉
holomorph	G ⋊ Aut(G) = #453×AGL(1,𝔽₁₉)
homology	Abelianization (H₁) = C₂²×C₁₉ · Schur multiplier (H₂) = C₂

Aliases

direct product	D₄×C₁₉
unambiguous semidirect product	C₄⋊(C₂×C₁₉)
unambiguous semidirect product	C₂²⋊(C₂×C₁₉)

Subgroups

	subgroup	G/H
center, socle	C₂×C₁₉	C₂²
commutator, Frattini, cosocle	C₂	C₂²×C₁₉
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₉ ◁ G

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 23
transitive degree	μ_t(G) = 76

A minimal faithful permutation representation — 2 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(152, 9);