C₃×D₂₁₅ #13046261

order	1290
isoclinism	isoclinic class of D₂₁₅
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₃×D₂₁₅
unambiguous semidirect product	C₅⋊(C₃×D₄₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₄₃ ⋊ C₂
derived, Hall	C₅×C₄₃ ⋊ C₂×C₃
Hall	C₃×C₄₃ ⋊ D₅
Sylow	C₄₃ ⋊ C₃×D₅
Hall	C₃×C₅ ⋊ D₄₃
Sylow	C₅ ⋊ C₃×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₃	D₂₁₅
commutator, cosocle, hypocenter	C₅×C₄₃	C₂×C₃
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₄₃	C₂
LCS	G ▷ C₅×C₄₃
UCS	C₁ ◁ C₃

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)(10,51)(11,50)(12,49)(13,48)(14,47)(15,46)(16,45)(17,44)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₃×D₅×D₄₃, D₃×D₂₁₅, D₁₅⋊D₄₃, …
maximal normal subgroup of	C₃×D₅×D₄₃, D₃×D₂₁₅, D₁₅⋊D₄₃, …
maximal quotient of	C₃×Dic₂₁₅, C₂×C₃×D₂₁₅, C₉×D₂₁₅, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1290, 7);

C3×D215 #13046261