#1043⋊C₃ #64481

order	300
isoclinism	isoclinic class of #1043⋊C₃
automorphism	Aut(G) = AΓL(1,𝔽₂₅) (order 1200) · Inn(G) = G · Out(G) = C₂²
homology	Abelianization (H₁) = C₃×C₄ · Schur multiplier (H₂) = C₁

Aliases

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, hypocenter	C₅²	C₃×C₄
prime radical	(C₅⋊D₅)⋊C₃	C₂
cosocle	C₅⋊D₅	C₂×C₃
LCS	G ▷ C₅²
UCS	C₁

18 conjugacy classes of subgroups.

minimal degree	μ(G) = 15
transitive degree	μ_t(G) = 15

A minimal faithful permutation representation — 2 generators on 15 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 24);