C₁₁×D₂₁₃ #17989759

order	4686
isoclinism	isoclinic class of D₂₁₃
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₁ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₁×D₂₁₃
unambiguous semidirect product	C₃⋊(C₁₁×D₇₁)
unambiguous semidirect product	C₇₁⋊(D₃×C₁₁)

Splittings

Fitting, socle, Hall	C₃×C₁₁×C₇₁ ⋊ C₂
derived, Hall	C₃×C₇₁ ⋊ C₂×C₁₁
Hall	C₁₁×C₇₁ ⋊ D₃
Sylow	C₇₁ ⋊ D₃×C₁₁
Hall	C₃×C₁₁ ⋊ D₇₁
Sylow	C₃ ⋊ C₁₁×D₇₁

Subgroups

	subgroup	G/H
center, hypercenter	C₁₁	D₂₁₃
commutator, cosocle, hypocenter	C₃×C₇₁	C₂×C₁₁
Frattini	C₁	G
Fitting, socle	C₃×C₁₁×C₇₁	C₂
LCS	G ▷ C₃×C₇₁
UCS	C₁ ◁ C₁₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 2343

A minimal faithful permutation representation — 2 generators on 85 points:

(1,2,3,4,5,6,7,8,9,10,11)(12,13,14)(15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(13,14)(16,85)(17,84)(18,83)(19,82)(20,81)(21,80)(22,79)(23,78)(24,77)(25,76)(26,75)(27,74)(28,73)(29,72)(30,71)(31,70)(32,69)(33,68)(34,67)(35,66)(36,65)(37,64)(38,63)(39,62)(40,61)(41,60)(42,59)(43,58)(44,57)(45,56)(46,55)(47,54)(48,53)(49,52)(50,51)

Conjugacy classes & character table

1188 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₁₁×D₆₃₉, C₁₁×C₃⋊D₂₁₃, C₃×C₁₁×D₂₁₃, …
maximal normal subgroup of	C₃×C₁₁×D₂₁₃, C₅×C₁₁×D₂₁₃, C₁₁×C₃⋊(D₇₁⋊C₅), …
maximal quotient of	C₁₁×D₆₃₉, C₁₁×C₃⋊D₂₁₃, C₃×C₁₁×D₂₁₃, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(4686, 6);

C11×D213 #17989759