C₃×D₅×C₄₃ #13046260

order	1290
isoclinism	isoclinic class of D₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃×C₄₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₅×C₄₃
unambiguous semidirect product	C₅⋊(C₂×C₃×C₄₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₄₃ ⋊ C₂
derived, Sylow	C₅ ⋊ C₂×C₃×C₄₃
Hall	C₅×C₄₃ ⋊ C₂×C₃
Hall	C₃×C₅ ⋊ C₂×C₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃×C₄₃	D₅
commutator, cosocle, hypocenter	C₅	C₂×C₃×C₄₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₄₃	C₂
LCS	G ▷ C₅
UCS	C₁ ◁ C₃×C₄₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 51
transitive degree	μ_t(G) = 645

A minimal faithful permutation representation — 2 generators on 51 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);(5,8)(6,7)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×AGL(1,𝔽₅)×C₄₃, C₃×C₄₃⋊AGL(1,𝔽₅), C₃×D₅×D₄₃, …
maximal normal subgroup of	C₃×AGL(1,𝔽₅)×C₄₃, C₃×C₄₃⋊AGL(1,𝔽₅), C₃×D₅×D₄₃, …
maximal quotient of	C₃×Dic₅×C₄₃, C₂×C₃×D₅×C₄₃, C₉×D₅×C₄₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1290, 6);

C3×D5×C43 #13046260