#3874⋊C₂ #64482

order	300
isoclinism	isoclinic class of #3874⋊C₂
automorphism	Aut(G) = AΓL(1,𝔽₂₅) (order 1200) · Inn(G) = G · Out(G) = C₄
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

unambiguous semidirect product	#3874⋊C₂
unambiguous semidirect product	((C₅⋊D₅)⋊C₃)⋊C₂

Fitting, socle, Sylow	C₅² ⋊ C₂×D₃
derived, Hall	C₅²⋊C₃ ⋊ C₂²

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅²⋊C₃	C₂²
Fitting, socle	C₅²	C₂×D₃
LCS	G ▷ C₅²⋊C₃
UCS	C₁

30 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:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(300, 25);