C₁₃×D₆₁ #13097536

order	1586
isoclinism	isoclinic class of D₆₁
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₆₁
unambiguous semidirect product	C₆₁⋊(C₂×C₁₃)

Splittings

Fitting, socle, Hall	C₁₃×C₆₁ ⋊ C₂
derived, Sylow	C₆₁ ⋊ C₂×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	D₆₁
commutator, cosocle, hypocenter	C₆₁	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₁₃×C₆₁	C₂
LCS	G ▷ C₆₁
UCS	C₁ ◁ C₁₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 74
transitive degree	μ_t(G) = 793

A minimal faithful permutation representation — 2 generators on 74 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);(15,74)(16,73)(17,72)(18,71)(19,70)(20,69)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)(27,62)(28,61)(29,60)(30,59)(31,58)(32,57)(33,56)(34,55)(35,54)(36,53)(37,52)(38,51)(39,50)(40,49)(41,48)(42,47)(43,46)(44,45)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	C₁₃×#6923, #15691761, D₁₃×D₆₁, …
maximal normal subgroup of	C₁₃×#6923, #15691761, D₁₃×D₆₁, …
maximal quotient of	C₁₃×Dic₆₁, C₂×C₁₃×D₆₁, C₃×C₁₃×D₆₁, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1586, 2);

C13×D61 #13097536