D₈₀₉₁ #30633644

order	16182
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₈₀₉₁
unambiguous semidirect product	C₉⋊D₈₉₉
unambiguous semidirect product	C₂₉⋊D₂₇₉
unambiguous semidirect product	C₃₁⋊D₂₆₁

Splittings

Fitting, derived, Hall	C₉×C₂₉×C₃₁ ⋊ C₂
Hall	C₂₉×C₃₁ ⋊ D₉
Hall	C₉×C₃₁ ⋊ D₂₉
Hall	C₉×C₂₉ ⋊ D₃₁
Sylow	C₃₁ ⋊ D₂₆₁
Sylow	C₂₉ ⋊ D₂₇₉
Sylow	C₉ ⋊ D₈₉₉

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	C₉×C₂₉×C₃₁	C₂
Frattini	C₃	D₂₆₉₇
socle	C₃×C₂₉×C₃₁	D₃
LCS	G ▷ C₉×C₂₉×C₃₁
UCS	C₁

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 69
transitive degree	μ_t(G) = 8091

A minimal faithful permutation representation — 2 generators on 69 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);(2,9)(3,8)(4,7)(5,6)(11,38)(12,37)(13,36)(14,35)(15,34)(16,33)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)(40,69)(41,68)(42,67)(43,66)(44,65)(45,64)(46,63)(47,62)(48,61)(49,60)(50,59)(51,58)(52,57)(53,56)(54,55)

Polycyclic presentation:

2,3,29,31,3|b^3=e^2,b^a=b^2e,c^a=c^28,d^a=d^30,e^a=e^2

Conjugacy classes & character table

4047 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₃₁×D₂₆₁, D₂₉×D₂₇₉, D₉×D₈₉₉, …
maximal normal subgroup of	D₃₁×D₂₆₁, D₂₉×D₂₇₉, D₉×D₈₉₉, …
maximal quotient of	Dic₈₀₉₁, C₂×D₈₀₉₁, D₂₄₂₇₃

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16182, 9);

D8091 #30633644