C₃⋊(C₁₉⋊C₈) #90685

order	456
isoclinism	isoclinic class of D₅₇
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₁

Aliases

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

	subgroup	G/H
center, Frattini, hypercenter	C₄	D₅₇
commutator, hypocenter	C₃×C₁₉	C₈
Fitting, cosocle	C₃×C₄×C₁₉	C₂
socle, prime radical	C₂×C₃×C₁₉	C₄
LCS	G ▷ C₃×C₁₉
UCS	C₁ ◁ C₄

16 conjugacy classes of subgroups.

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 456

A minimal faithful permutation representation — 2 generators on 30 points:

(2,3)(4,5)(6,7)(8,9)(10,11)(12,13)(14,15)(16,17)(18,19)(21,22)(23,24,25,27,26,28,29,30);(1,2,4,6,8,10,12,14,16,18,19,17,15,13,11,9,7,5,3)(20,21,22)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(456, 5);