C₅⋊PSU(3,𝔽₂) #67206

order	360
isoclinism	isoclinic class of C₅⋊PSU(3,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₃

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₅×C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃²×C₅	Q₈
LCS	G ▷ C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁

28 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 45

A minimal faithful permutation representation — 2 generators on 14 points:

(1,7,6,8)(3,4,9,5)(10,11,13,14,12);(2,4,6,8)(3,7,5,9)(11,12)(13,14)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 136);