D₆₇₇₁ #27214367

order	13542
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₇₇₁
unambiguous semidirect product	C₃⋊D₂₂₅₇
unambiguous semidirect product	C₃₇⋊D₁₈₃
unambiguous semidirect product	C₆₁⋊D₁₁₁

Splittings

Fitting, socle, derived, Hall	C₃×C₃₇×C₆₁ ⋊ C₂
Hall	C₃₇×C₆₁ ⋊ D₃
Hall	C₃×C₆₁ ⋊ D₃₇
Hall	C₃×C₃₇ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₁₁₁
Sylow	C₃₇ ⋊ D₁₈₃
Sylow	C₃ ⋊ D₂₂₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃×C₃₇×C₆₁	C₂
LCS	G ▷ C₃×C₃₇×C₆₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 6771

A minimal faithful permutation representation — 2 generators on 101 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)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101);(2,3)(5,40)(6,39)(7,38)(8,37)(9,36)(10,35)(11,34)(12,33)(13,32)(14,31)(15,30)(16,29)(17,28)(18,27)(19,26)(20,25)(21,24)(22,23)(42,101)(43,100)(44,99)(45,98)(46,97)(47,96)(48,95)(49,94)(50,93)(51,92)(52,91)(53,90)(54,89)(55,88)(56,87)(57,86)(58,85)(59,84)(60,83)(61,82)(62,81)(63,80)(64,79)(65,78)(66,77)(67,76)(68,75)(69,74)(70,73)(71,72)

Polycyclic presentation:

2,3,37,61|b^a=b^2,c^a=c^36,d^a=d^60

Conjugacy classes & character table

3387 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₆₁×D₁₁₁, D₃₇×D₁₈₃, D₃×D₂₂₅₇, …
maximal normal subgroup of	D₆₁×D₁₁₁, D₃₇×D₁₈₃, D₃×D₂₂₅₇, …
maximal quotient of	Dic₆₇₇₁, C₂×D₆₇₇₁

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13542, 12);

D6771 #27214367