D₃²⋊D₅ #67203

order	360
isoclinism	isoclinic class of D₃²⋊D₅
automorphism	Aut(G) = AGL(1,𝔽₅)×GO⁺(4,𝔽₂) (order 1440) · Inn(G) = G · Out(G) = C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

D₃²⋊D₅

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₅×C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃²×C₅	D₄
LCS	G ▷ C₅×C₃⋊D₃ ▷ C₃²×C₅
UCS	C₁

52 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(360, 133);