C₁₃×D₁₂₉ #16809804

order	3354
isoclinism	isoclinic class of D₁₂₉
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₁₂₉
unambiguous semidirect product	C₃⋊(C₁₃×D₄₃)
unambiguous semidirect product	C₄₃⋊(D₃×C₁₃)

Splittings

Fitting, socle, Hall	C₃×C₁₃×C₄₃ ⋊ C₂
derived, Hall	C₃×C₄₃ ⋊ C₂×C₁₃
Hall	C₁₃×C₄₃ ⋊ D₃
Sylow	C₄₃ ⋊ D₃×C₁₃
Hall	C₃×C₁₃ ⋊ D₄₃
Sylow	C₃ ⋊ C₁₃×D₄₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	D₁₂₉
commutator, cosocle, hypocenter	C₃×C₄₃	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₃×C₁₃×C₄₃	C₂
LCS	G ▷ C₃×C₄₃
UCS	C₁ ◁ C₁₃

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 59
transitive degree	μ_t(G) = 1677

A minimal faithful permutation representation — 2 generators on 59 points:

(1,2,3,4,5,6,7,8,9,10,11,12,13)(14,15,16)(17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59);(15,16)(18,59)(19,58)(20,57)(21,56)(22,55)(23,54)(24,53)(25,52)(26,51)(27,50)(28,49)(29,48)(30,47)(31,46)(32,45)(33,44)(34,43)(35,42)(36,41)(37,40)(38,39)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₅×C₁₃×D₁₂₉, C₁₃×D₆₄₅, C₇×C₁₃×D₁₂₉, …
maximal normal subgroup of	C₅×C₁₃×D₁₂₉, C₇×C₁₃×D₁₂₉, C₁₃×C₃⋊(D₄₃⋊C₇), …
maximal quotient of	C₅×C₁₃×D₁₂₉, C₁₃×D₆₄₅, C₇×C₁₃×D₁₂₉, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(3354, 10);

C13×D129 #16809804