Dic₄₅₆₅ #32912416

order	18260
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₆₅
unambiguous semidirect product	C₁₁⋊Dic₄₁₅
unambiguous semidirect product	C₈₃⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₈₃ ⋊ C₄
Hall	C₁₁×C₈₃ ⋊ Dic₅
Hall	C₅×C₈₃ ⋊ Dic₁₁
Sylow	C₈₃ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₈₃
Sylow	C₁₁ ⋊ Dic₄₁₅
Sylow	C₅ ⋊ Dic₉₁₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₄₅₆₅
commutator, hypocenter	C₅×C₁₁×C₈₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₈₃	C₂
LCS	G ▷ C₅×C₁₁×C₈₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 103
transitive degree	μ_t(G) = 18260

A minimal faithful permutation representation — 2 generators on 103 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);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,103)(23,102)(24,101)(25,100)(26,99)(27,98)(28,97)(29,96)(30,95)(31,94)(32,93)(33,92)(34,91)(35,90)(36,89)(37,88)(38,87)(39,86)(40,85)(41,84)(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:

2,5,11,83,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^82

Conjugacy classes & character table

4568 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₁₃₀, Dic₁₃₆₉₅
maximal normal subgroup of	Dic₉₁₃₀
maximal quotient of	Dic₁₃₆₉₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18260, 9);

Dic4565 #32912416