C₂²×D₁₉ #3894

order	152
isoclinism	isoclinic class of D₁₉
automorphism	Aut(G) = S₄×AGL(1,𝔽₁₉) (order 8208) · Inn(G) = D₁₉ · Out(G) = C₉×S₄
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

direct product	C₂²×D₁₉
unambiguous semidirect product	C₁₉⋊C₂³
unambiguous semidirect product	(C₂×C₁₉)⋊C₂²

Fitting, socle	C₂²×C₁₉ ⋊ C₂
derived, Sylow	C₁₉ ⋊ C₂³

	subgroup	G/H
center, hypercenter	C₂²	D₁₉
commutator, cosocle, hypocenter	C₁₉	C₂³
Frattini	C₁	G
Fitting, socle	C₂²×C₁₉	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂²

32 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 3 generators on 23 points:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(152, 11);