D₆₅₆₅ #27190052

order	13130
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₁

Aliases

dihedral	D₆₅₆₅
unambiguous semidirect product	C₅⋊D₁₃₁₃
unambiguous semidirect product	C₁₃⋊D₅₀₅
unambiguous semidirect product	C₁₀₁⋊D₆₅

Splittings

Fitting, socle, derived, Hall	C₅×C₁₃×C₁₀₁ ⋊ C₂
Hall	C₁₃×C₁₀₁ ⋊ D₅
Hall	C₅×C₁₀₁ ⋊ D₁₃
Sylow	C₁₀₁ ⋊ D₆₅
Hall	C₅×C₁₃ ⋊ D₁₀₁
Sylow	C₁₃ ⋊ D₅₀₅
Sylow	C₅ ⋊ D₁₃₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₅×C₁₃×C₁₀₁	C₂
LCS	G ▷ C₅×C₁₃×C₁₀₁
UCS	C₁

16 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 6565

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

Polycyclic presentation:

2,5,13,101|b^a=b^4,c^a=c^12,d^a=d^100

Conjugacy classes & character table

3284 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#39910982, #39910983, #39910986, …
maximal normal subgroup of	#39910982, #39910983, #39910986, …
maximal quotient of	Dic₆₅₆₅, C₂×D₆₅₆₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13130, 9);

D6565 #27190052