A₄⋊D₁₉ #90723

order	456
isoclinism	isoclinic class of A₄⋊D₁₉
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	A₄⋊D₁₉
unambiguous semidirect product	C₁₉⋊S₄

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	A₄×C₁₉	C₂
Fitting, socle	C₂²×C₁₉	D₃
LCS	G ▷ A₄×C₁₉
UCS	C₁

22 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 43);