D₁₆₁₆₅ #44568258

order	32330
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₁₆₁₆₅
unambiguous semidirect product	C₅⋊D₃₂₃₃
unambiguous semidirect product	C₅₃⋊D₃₀₅
unambiguous semidirect product	C₆₁⋊D₂₆₅

Splittings

Fitting, socle, derived, Hall	C₅×C₅₃×C₆₁ ⋊ C₂
Hall	C₅₃×C₆₁ ⋊ D₅
Hall	C₅×C₆₁ ⋊ D₅₃
Hall	C₅×C₅₃ ⋊ D₆₁
Sylow	C₆₁ ⋊ D₂₆₅
Sylow	C₅₃ ⋊ D₃₀₅
Sylow	C₅ ⋊ D₃₂₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₅₃×C₆₁	C₂
LCS	G ▷ C₅×C₅₃×C₆₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 16165

A minimal faithful permutation representation — 2 generators on 119 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)(59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119);(2,5)(3,4)(7,58)(8,57)(9,56)(10,55)(11,54)(12,53)(13,52)(14,51)(15,50)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(22,43)(23,42)(24,41)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,34)(32,33)(60,119)(61,118)(62,117)(63,116)(64,115)(65,114)(66,113)(67,112)(68,111)(69,110)(70,109)(71,108)(72,107)(73,106)(74,105)(75,104)(76,103)(77,102)(78,101)(79,100)(80,99)(81,98)(82,97)(83,96)(84,95)(85,94)(86,93)(87,92)(88,91)(89,90)

Polycyclic presentation:

2,5,53,61|b^a=b^4,c^a=c^52,d^a=d^60

Conjugacy classes & character table

8084 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₂×D₁₆₁₆₅
maximal normal subgroup of	C₂×D₁₆₁₆₅
maximal quotient of	Dic₁₆₁₆₅, C₂×D₁₆₁₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(32330, 9);

D16165 #44568258