C₃⋊(C₃⋊#170) #67124

order	360
isoclinism	isoclinic class of C₃⋊ΓL(1,𝔽₁₆)
automorphism	?
homology	Abelianization (H₁) = C₈ · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₂	C₃⋊ΓL(1,𝔽₁₆)
commutator, hypocenter	C₃²×C₅	C₈
Fitting, socle, prime radical	C₂×C₃²×C₅	C₄
cosocle	C₃²×Dic₅	C₂
LCS	G ▷ C₃²×C₅
UCS	C₁ ◁ C₂

48 conjugacy classes of subgroups.

minimal degree	μ(G) = 19
transitive degree	μ_t(G) = 360

A minimal faithful permutation representation — 3 generators on 19 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 54);