C₄₁×D₅₅ #17985231

order	4510
isoclinism	isoclinic class of D₅₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₁×D₅₅
unambiguous semidirect product	C₅⋊(D₁₁×C₄₁)
unambiguous semidirect product	C₁₁⋊(D₅×C₄₁)

Splittings

Fitting, socle, Hall	C₅×C₁₁×C₄₁ ⋊ C₂
derived, Hall	C₅×C₁₁ ⋊ C₂×C₄₁
Hall	C₁₁×C₄₁ ⋊ D₅
Hall	C₅×C₄₁ ⋊ D₁₁
Sylow	C₁₁ ⋊ D₅×C₄₁
Sylow	C₅ ⋊ D₁₁×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	D₅₅
commutator, cosocle, hypocenter	C₅×C₁₁	C₂×C₄₁
Frattini	C₁	G
Fitting, socle	C₅×C₁₁×C₄₁	C₂
LCS	G ▷ C₅×C₁₁
UCS	C₁ ◁ 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);(43,46)(44,45)(48,57)(49,56)(50,55)(51,54)(52,53)

Conjugacy classes & character table

1189 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃×C₄₁×D₅₅, C₄₁×D₁₆₅, C₄₁×D₂₇₅, …
maximal normal subgroup of	C₃×C₄₁×D₅₅, C₄₁×C₅⋊AGL(1,𝔽₁₁), C₅⋊#17985241, …
maximal quotient of	C₃×C₄₁×D₅₅, C₄₁×D₁₆₅, C₄₁×D₂₇₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4510, 13);

C41×D55 #17985231