D₁₉⋊A₄ #90726

order	456
isoclinism	isoclinic class of D₁₉⋊A₄
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	D₁₉⋊A₄
unambiguous semidirect product	(C₂²⋊(C₁₉⋊C₃))⋊C₂

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₁₉	C₂×C₃
LCS	G ▷ C₂²×C₁₉
UCS	C₁

24 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)(21,22,23);(1,2,5,11,17,14,18,4,8,13,19,7,6,12,10,15,16,9,3)(20,23)(21,22)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 46);