D₅⋊#170 #87833

order	400
isoclinism	isoclinic class of D₅⋊AGL(1,𝔽₅)
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)² (order 1600) · Inn(G) = D₅⋊AGL(1,𝔽₅) · Out(G) = C₂³
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

C₅² ⋊ C₂×C₈

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅⋊AGL(1,𝔽₅)
commutator, hypocenter	C₅²	C₂×C₈
Fitting, socle	C₂×C₅²	C₂×C₄
prime radical	C₂×C₅×D₅	C₄
cosocle	#1039	C₂²
LCS	G ▷ C₅²
UCS	C₁ ◁ C₂

52 conjugacy classes of subgroups.

minimal degree	μ(G) = 18
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 2 generators on 18 points:

(9,10)(12,13)(14,15,18,16,17);(1,2,3,5,4,6,7,8)(10,11,12,13)(15,16,17,18)

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 127);