ΓL(1,𝔽₁₆)⋊D₃ #67199

order	360
isoclinism	isoclinic class of ΓL(1,𝔽₁₆)⋊D₃
automorphism	Aut(G) = AGL(1,𝔽₅)×GO⁺(4,𝔽₂) (order 1440) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₃²×C₅	C₂×C₄
prime radical	D₅×C₃⋊D₃	C₂
cosocle	C₃²×D₅	C₂²
LCS	G ▷ C₃²×C₅
UCS	C₁

74 conjugacy classes of subgroups.

minimal degree	μ(G) = 11
transitive degree	μ_t(G) = 30

A minimal faithful permutation representation — 2 generators on 11 points:

(1,2)(5,6)(7,8,11,9,10);(2,3)(4,5,6)(8,9,10,11)

Polycyclic presentation:

2^3,3^2,5|a^2=c,d^a=d^2,f^a=f^3,d^b=d^2,e^b=e^2,f^c=f^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 129);