(C₁₉⋊C₃)⋊C₈ #90681

order	456
isoclinism	isoclinic class of D₁₉⋊C₃
automorphism	Aut(G) = C₂²×AGL(1,𝔽₁₉) (order 1368) · Inn(G) = D₁₉⋊C₃ · Out(G) = C₂²×C₃
homology	Abelianization (H₁) = C₃×C₈ · Schur multiplier (H₂) = C₁

Aliases

unambiguous semidirect product	(C₁₉⋊C₃)⋊C₈
unambiguous semidirect product	(C₁₉⋊C₈)⋊C₃

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₁₉⋊C₃
commutator, hypocenter	C₁₉	C₃×C₈
Fitting, cosocle	C₄×C₁₉	C₂×C₃
socle	C₂×C₁₉	C₃×C₄
prime radical	C₂×C₁₉⋊C₃	C₄
LCS	G ▷ C₁₉
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 27
transitive degree	μ_t(G) = 152

A minimal faithful permutation representation — 2 generators on 27 points:

(1,2,3,5,4,6,7,8)(10,14,15,11,12,16)(13,22,19,17,18,24)(20,25,27,26,23,21);(9,10,13,19,25,22,26,12,16,21,27,15,14,20,18,23,24,17,11)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 1);