(D₅⋊AGL(1,𝔽₅))⋊C₂ #87913

order	400
isoclinism	isoclinic class of (D₅⋊AGL(1,𝔽₅))⋊C₂
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

unambiguous semidirect product	(D₅⋊AGL(1,𝔽₅))⋊C₂
unambiguous semidirect product	(D₅²⋊C₂)⋊C₂
unambiguous semidirect product	#6057⋊C₂

C₅² ⋊ #41

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle	C₅⋊D₅	C₂³
Fitting, socle, hypocenter	C₅²	#41
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁

64 conjugacy classes of subgroups.

minimal degree	μ(G) = 10
transitive degree	μ_t(G) = 10

A minimal faithful permutation representation — 3 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 207);