C₄₁×D₄₅ #16814799

order	3690
isoclinism	isoclinic class of D₄₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₄₁×D₄₅
unambiguous semidirect product	C₅⋊(D₉×C₄₁)
unambiguous semidirect product	C₉⋊(D₅×C₄₁)

Splittings

Fitting, Hall	C₅×C₉×C₄₁ ⋊ C₂
derived, Hall	C₅×C₉ ⋊ C₂×C₄₁
Hall	C₉×C₄₁ ⋊ D₅
Hall	C₅×C₄₁ ⋊ D₉
Sylow	C₉ ⋊ D₅×C₄₁
Sylow	C₅ ⋊ D₉×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	D₄₅
commutator, cosocle, hypocenter	C₅×C₉	C₂×C₄₁
Frattini	C₃	D₁₅×C₄₁
Fitting	C₅×C₉×C₄₁	C₂
socle	C₃×C₅×C₄₁	D₃
LCS	G ▷ C₅×C₉
UCS	C₁ ◁ C₄₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1845

A minimal faithful permutation representation — 2 generators on 55 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);(43,50)(44,49)(45,48)(46,47)(52,55)(53,54)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₄₁×D₂₂₅, D₄₅×C₄₁⋊C₅, C₅×C₄₁×D₄₅, …
maximal normal subgroup of	D₄₅×C₄₁⋊C₅, C₅×C₄₁×D₄₅, C₇×C₄₁×D₄₅
maximal quotient of	C₄₁×D₂₂₅, C₅×C₄₁×D₄₅, C₄₁×C₅⋊D₄₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3690, 11);

C41×D45 #16814799