C₂×C₄₁×C₁₁⋊C₅ #17985221

order	4510
isoclinism	isoclinic class of C₁₁⋊C₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₅×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

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

Splittings

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

Subgroups

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

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 54
transitive degree	μ_t(G) = 902

A minimal faithful permutation representation — 2 generators on 54 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)(45,46,48,51,50)(47,49,53,54,52);(44,45,47,50,46,48,52,54,49,51,53)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₂×C₃×C₄₁×C₁₁⋊C₅, D₃×C₄₁×C₁₁⋊C₅, C₂×C₁₁⋊C₅×C₄₁⋊C₅, …
maximal normal subgroup of	C₂×C₃×C₄₁×C₁₁⋊C₅, C₂×C₁₁⋊C₅×C₄₁⋊C₅, C₂×C₅×C₄₁×C₁₁⋊C₅, …
maximal quotient of	C₂×C₃×C₄₁×C₁₁⋊C₅, D₃×C₄₁×C₁₁⋊C₅, C₂×C₄₁×C₁₁⋊C₂₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4510, 3);

C2×C41×C11⋊C5 #17985221