C₁₃×D₆₇ #13181985

order	1742
isoclinism	isoclinic class of D₆₇
automorphism	?
homology	Abelianization (H₁) = C₂×C₁₃ · Schur multiplier (H₂) = C₁

Aliases

direct product	C₁₃×D₆₇
unambiguous semidirect product	C₆₇⋊(C₂×C₁₃)

Splittings

Fitting, socle, Hall	C₁₃×C₆₇ ⋊ C₂
derived, Sylow	C₆₇ ⋊ C₂×C₁₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁₃	D₆₇
commutator, cosocle, hypocenter	C₆₇	C₂×C₁₃
Frattini	C₁	G
Fitting, socle	C₁₃×C₆₇	C₂
LCS	G ▷ C₆₇
UCS	C₁ ◁ C₁₃

8 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 871

A minimal faithful permutation representation — 2 generators on 80 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);(15,80)(16,79)(17,78)(18,77)(19,76)(20,75)(21,74)(22,73)(23,72)(24,71)(25,70)(26,69)(27,68)(28,67)(29,66)(30,65)(31,64)(32,63)(33,62)(34,61)(35,60)(36,59)(37,58)(38,57)(39,56)(40,55)(41,54)(42,53)(43,52)(44,51)(45,50)(46,49)(47,48)

Conjugacy classes & character table

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

Relationships

maximal subgroup of	D₁₃×D₆₇, C₂×C₁₃×D₆₇, C₇×C₁₃×D₆₇, …
maximal normal subgroup of	D₁₃×D₆₇, C₂×C₁₃×D₆₇, C₇×C₁₃×D₆₇, …
maximal quotient of	C₁₃×Dic₆₇, C₂×C₁₃×D₆₇, C₇×C₁₃×D₆₇, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1742, 2);

C13×D67 #13181985