C₂²⋊(C₁₉⋊C₃) #6585

order	228
isoclinism	isoclinic class of C₂²⋊(C₁₉⋊C₃)
automorphism	Aut(G) = A₄×AGL(1,𝔽₁₉) (order 4104) · Inn(G) = G · Out(G) = C₂×C₉
homology	Abelianization (H₁) = C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	C₂²⋊(C₁₉⋊C₃)
unambiguous semidirect product	C₁₉⋊A₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₁₉	C₃
LCS	G ▷ C₂²×C₁₉
UCS	C₁

10 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(228, 11);