C₁₁⋊C₂₅ #63275

order	275
isoclinism	isoclinic class of C₁₁⋊C₅
automorphism	Aut(G) = C₅×AGL(1,𝔽₁₁) (order 550) · Inn(G) = C₁₁⋊C₅ · Out(G) = C₂×C₅
homology	Abelianization (H₁) = C₂₅ · Schur multiplier (H₂) = C₁

Aliases

C₁₁⋊C₂₅

C₁₁ ⋊ C₂₅

	subgroup	G/H
center, Frattini, hypercenter	C₅	C₁₁⋊C₅
commutator, hypocenter	C₁₁	C₂₅
Fitting, socle, prime radical, cosocle	C₅×C₁₁	C₅
LCS	G ▷ C₁₁
UCS	C₁ ◁ C₅

6 conjugacy classes of subgroups.

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 275

A minimal faithful permutation representation — 2 generators on 36 points:

(2,3,5,8,7)(4,6,10,11,9)(12,13,15,18,22,14,16,19,23,27,17,20,24,28,31,21,25,29,32,34,26,30,33,35,36);(1,2,4,7,3,5,9,11,6,8,10)

Polycyclic presentation:

5^2,11|a^5=b,c^a=c^4

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(275, 1);