C₃×C₅×D₄₁ #11927609

order	1230
isoclinism	isoclinic class of D₄₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product

C₃×C₅×D₄₁

Splittings

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

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₅	D₄₁
commutator, cosocle, hypocenter	C₄₁	C₂×C₃×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₄₁	C₂
LCS	G ▷ C₄₁
UCS	C₁ ◁ C₃×C₅

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);(10,49)(11,48)(12,47)(13,46)(14,45)(15,44)(16,43)(17,42)(18,41)(19,40)(20,39)(21,38)(22,37)(23,36)(24,35)(25,34)(26,33)(27,32)(28,31)(29,30)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×C₅×#4223, C₃×#11717964, C₃×D₅×D₄₁, …
maximal normal subgroup of	C₃×C₅×#4223, C₃×#11717964, C₃×D₅×D₄₁, …
maximal quotient of	C₃×C₅×Dic₄₁, C₂×C₃×C₅×D₄₁, C₅×C₉×D₄₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1230, 5);

C3×C5×D41 #11927609