(C₂×AGL(1,𝔽₅))⋊D₅ #87824

order	400
isoclinism	isoclinic class of (C₂×AGL(1,𝔽₅))⋊D₅
automorphism	Aut(G) = C₂²×AGL(1,𝔽₅)² (order 1600) · Inn(G) = D₅×AGL(1,𝔽₅) · Out(G) = C₂×C₄
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₂²

Aliases

C₅² ⋊ C₂²⋊C₄

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅×AGL(1,𝔽₅)
commutator, Fitting, socle	C₂×C₅²	C₂×C₄
prime radical	C₂×D₅²	C₂
cosocle	C₂×C₅×D₅	C₂²
hypocenter	C₅²	C₂²⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂

73 conjugacy classes of subgroups.

minimal degree	μ(G) = 14
transitive degree	μ_t(G) = 40

A minimal faithful permutation representation — 2 generators on 14 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 118);