C₅×D₁₂₉ #13046263

order	1290
isoclinism	isoclinic class of D₁₂₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₅ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₅×D₁₂₉
unambiguous semidirect product	C₃⋊(C₅×D₄₃)
unambiguous semidirect product	C₄₃⋊(C₅×D₃)

Splittings

Fitting, socle, Hall	C₃×C₅×C₄₃ ⋊ C₂
derived, Hall	C₃×C₄₃ ⋊ C₂×C₅
Hall	C₅×C₄₃ ⋊ D₃
Sylow	C₄₃ ⋊ C₅×D₃
Hall	C₃×C₅ ⋊ D₄₃
Sylow	C₃ ⋊ C₅×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₅	D₁₂₉
commutator, cosocle, hypocenter	C₃×C₄₃	C₂×C₅
Frattini	C₁	G
Fitting, socle	C₃×C₅×C₄₃	C₂
LCS	G ▷ C₃×C₄₃
UCS	C₁ ◁ C₅

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);(7,8)(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

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

Relationships

maximal subgroup of	C₅×D₃×D₄₃, D₅×D₁₂₉, D₁₅⋊D₄₃, …
maximal normal subgroup of	C₅×D₃×D₄₃, D₅×D₁₂₉, D₁₅⋊D₄₃, …
maximal quotient of	C₅×Dic₁₂₉, C₂×C₅×D₁₂₉, C₅×D₃₈₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1290, 9);

C5×D129 #13046263