#6053⋊C₂ #87912

order	400
isoclinism	isoclinic class of #6053⋊C₂
automorphism	Aut(G) = AGL(1,𝔽₅)²⋊C₂ (order 800) · Inn(G) = G · Out(G) = C₂
homology	Abelianization (H₁) = C₂×C₄ · Schur multiplier (H₂) = C₁

Aliases

C₅² ⋊ M₁₆

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator	C₅⋊D₅	C₂×C₄
Fitting, socle, hypocenter	C₅²	M₁₆
prime radical	D₅²	C₄
cosocle	#1043	C₂²
LCS	G ▷ C₅⋊D₅ ▷ C₅²
UCS	C₁

31 conjugacy classes of subgroups.

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

A minimal faithful permutation representation — 2 generators on 10 points:

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

Polycyclic presentation:

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

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(400, 206);