D₅×C₁₁×C₄₁ #17985229

order	4510
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	D₅×C₁₁×C₄₁
unambiguous semidirect product	C₅⋊(C₂×C₁₁×C₄₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₄₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₁₁×C₄₁
Hall	C₅×C₄₁ ⋊ C₂×C₁₁
Hall	C₅×C₁₁ ⋊ C₂×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁×C₄₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₁₁×C₄₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₄₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₁₁×C₄₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 57
transitive degree	μ_t(G) = 2255

A minimal faithful permutation representation — 2 generators on 57 points:

(1,2,3,4,5)(6,7,8,9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57);(2,5)(3,4)

Conjugacy classes & character table

1804 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×D₅×C₁₁×C₄₁, C₁₁×D₁₅×C₄₁, C₁₁×C₄₁×D₂₅, …
maximal normal subgroup of	C₃×D₅×C₁₁×C₄₁, D₅×C₁₁×C₄₁⋊C₅, D₅×C₄₁×C₁₁⋊C₅, …
maximal quotient of	C₃×D₅×C₁₁×C₄₁, C₁₁×D₁₅×C₄₁, C₁₁×C₄₁×D₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4510, 11);

D5×C11×C41 #17985229