(C₁₉⋊C₃)⋊C₄ #6575

order	228
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	Aut(G) = C₂×AGL(1,𝔽₁₉) (order 684) · Inn(G) = D₁₉⋊C₃ · Out(G) = C₂×C₃
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	(C₁₉⋊C₃)⋊C₄
unambiguous semidirect product	Dic₁₉⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₉⋊C₃
commutator, hypocenter	C₁₉	C₃×C₄
Fitting, socle, cosocle	C₂×C₁₉	C₂×C₃
prime radical	C₂×C₁₉⋊C₃	C₂
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₂

12 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:

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

Polycyclic presentation:

2,3,2,19|a^2=c,d^a=d^18,d^b=d^7

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(228, 1);