C₄₁×C₅²⋊C₃ #15690481

order	3075
isoclinism	isoclinic class of C₅²⋊C₃
automorphism	?
homology	Abelianization (H₁) = C₃×C₄₁ · Schur multiplier (H₂) = C₅

Aliases

direct product	C₄₁×C₅²⋊C₃
unambiguous semidirect product	C₅²⋊(C₃×C₄₁)
unambiguous semidirect product	(C₅²×C₄₁)⋊C₃

Splittings

Fitting, socle, Hall	C₅²×C₄₁ ⋊ C₃
derived, Sylow	C₅² ⋊ C₃×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₄₁	C₅²⋊C₃
commutator, cosocle, hypocenter	C₅²	C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₅²×C₄₁	C₃
LCS	G ▷ C₅²
UCS	C₁ ◁ C₄₁

12 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 615

A minimal faithful permutation representation — 2 generators on 56 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)(44,48,50)(46,51,54)(47,53,56)(49,52,55);(42,44,49,54,56)(43,48,52,46,47)(45,51,50,53,55)

Conjugacy classes & character table

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

Relationships

derived subgroup of	C₃×C₄₁⋊#3874, C₃×#27134196, C₃×#27134237, …
maximal subgroup of	C₄₁×C₂²⋊(C₅²⋊C₃), C₇×C₄₁×C₅²⋊C₃, C₄₁×C₇⋊(C₅²⋊C₃), …
maximal normal subgroup of	C₇×C₄₁×C₅²⋊C₃, C₁₁×C₄₁×C₅²⋊C₃
maximal quotient of	C₄₁×C₂²⋊(C₅²⋊C₃), C₇×C₄₁×C₅²⋊C₃, C₄₁×C₇⋊(C₅²⋊C₃), …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3075, 5);

C41×C52⋊C3 #15690481