C₃×D₅×C₄₁ #11927610

order	1230
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅×C₄₁
unambiguous semidirect product	C₅⋊(C₂×C₃×C₄₁)

Splittings

Fitting, socle, Hall	C₃×C₅×C₄₁ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃×C₄₁
Hall	C₅×C₄₁ ⋊ C₂×C₃
Hall	C₃×C₅ ⋊ C₂×C₄₁

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄₁	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₄₁
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₄₁	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₄₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 49
transitive degree	μ_t(G) = 615

A minimal faithful permutation representation — 2 generators on 49 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);(5,8)(6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₅)×C₄₁, C₃×#11717963, C₃×D₅×D₄₁, …
maximal normal subgroup of	C₃×AGL(1,𝔽₅)×C₄₁, C₃×#11717963, C₃×D₅×D₄₁, …
maximal quotient of	C₃×Dic₅×C₄₁, C₂×C₃×D₅×C₄₁, C₉×D₅×C₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1230, 6);

C3×D5×C41 #11927610