D₁₆₁₆₉ #44568297

order	32338
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₁₆₉
unambiguous semidirect product	C₁₉⋊D₈₅₁
unambiguous semidirect product	C₂₃⋊D₇₀₃
unambiguous semidirect product	C₃₇⋊D₄₃₇

Splittings

Fitting, socle, derived, Hall	C₁₉×C₂₃×C₃₇ ⋊ C₂
Hall	C₂₃×C₃₇ ⋊ D₁₉
Hall	C₁₉×C₃₇ ⋊ D₂₃
Hall	C₁₉×C₂₃ ⋊ D₃₇
Sylow	C₃₇ ⋊ D₄₃₇
Sylow	C₂₃ ⋊ D₇₀₃
Sylow	C₁₉ ⋊ D₈₅₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₁₉×C₂₃×C₃₇	C₂
LCS	G ▷ C₁₉×C₂₃×C₃₇
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 79
transitive degree	μ_t(G) = 16169

A minimal faithful permutation representation — 2 generators on 79 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);(2,19)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(44,79)(45,78)(46,77)(47,76)(48,75)(49,74)(50,73)(51,72)(52,71)(53,70)(54,69)(55,68)(56,67)(57,66)(58,65)(59,64)(60,63)(61,62)

Polycyclic presentation:

2,19,23,37|b^a=b^18,c^a=c^22,d^a=d^36

Conjugacy classes & character table

8086 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₁₆₉
maximal normal subgroup of	C₂×D₁₆₁₆₉
maximal quotient of	Dic₁₆₁₆₉, C₂×D₁₆₁₆₉

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32338, 8);

D16169 #44568297