Dic₆₆₆₅ #40998615

order	26660
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₆₆₆₅
unambiguous semidirect product	C₃₁⋊Dic₂₁₅
unambiguous semidirect product	C₄₃⋊Dic₁₅₅

Splittings

derived, Hall	C₅×C₃₁×C₄₃ ⋊ C₄
Hall	C₃₁×C₄₃ ⋊ Dic₅
Hall	C₅×C₄₃ ⋊ Dic₃₁
Hall	C₅×C₃₁ ⋊ Dic₄₃
Sylow	C₄₃ ⋊ Dic₁₅₅
Sylow	C₃₁ ⋊ Dic₂₁₅
Sylow	C₅ ⋊ Dic₁₃₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₆₆₆₅
commutator, hypocenter	C₅×C₃₁×C₄₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₃₁×C₄₃	C₂
LCS	G ▷ C₅×C₃₁×C₄₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 83
transitive degree	μ_t(G) = 26660

A minimal faithful permutation representation — 2 generators on 83 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);(1,3,2,4)(6,9)(7,8)(11,40)(12,39)(13,38)(14,37)(15,36)(16,35)(17,34)(18,33)(19,32)(20,31)(21,30)(22,29)(23,28)(24,27)(25,26)(42,83)(43,82)(44,81)(45,80)(46,79)(47,78)(48,77)(49,76)(50,75)(51,74)(52,73)(53,72)(54,71)(55,70)(56,69)(57,68)(58,67)(59,66)(60,65)(61,64)(62,63)

Polycyclic presentation:

2,5,31,43,2|a^2=e,b^a=b^4,c^a=c^30,d^a=d^42

Conjugacy classes & character table

6668 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₃₃₃₀
maximal normal subgroup of	Dic₁₃₃₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(26660, 9);

Dic6665 #40998615