D₁₆₀₂₇ #44545363

order	32054
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₀₂₇
unambiguous semidirect product	C₁₁⋊D₁₄₅₇
unambiguous semidirect product	C₃₁⋊D₅₁₇
unambiguous semidirect product	C₄₇⋊D₃₄₁

Splittings

Fitting, socle, derived, Hall	C₁₁×C₃₁×C₄₇ ⋊ C₂
Hall	C₃₁×C₄₇ ⋊ D₁₁
Hall	C₁₁×C₄₇ ⋊ D₃₁
Hall	C₁₁×C₃₁ ⋊ D₄₇
Sylow	C₄₇ ⋊ D₃₄₁
Sylow	C₃₁ ⋊ D₅₁₇
Sylow	C₁₁ ⋊ D₁₄₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₁×C₃₁×C₄₇	C₂
LCS	G ▷ C₁₁×C₃₁×C₄₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 16027

A minimal faithful permutation representation — 2 generators on 89 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);(2,11)(3,10)(4,9)(5,8)(6,7)(13,42)(14,41)(15,40)(16,39)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(44,89)(45,88)(46,87)(47,86)(48,85)(49,84)(50,83)(51,82)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2,11,31,47|b^a=b^10,c^a=c^30,d^a=d^46

Conjugacy classes & character table

8015 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₀₂₇
maximal normal subgroup of	C₂×D₁₆₀₂₇
maximal quotient of	Dic₁₆₀₂₇, C₂×D₁₆₀₂₇

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32054, 8);

D16027 #44545363