Dic₅₆₆₅ #36383602

order	22660
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₅₆₆₅
unambiguous semidirect product	C₁₁⋊Dic₅₁₅
unambiguous semidirect product	C₁₀₃⋊Dic₅₅

Splittings

derived, Hall	C₅×C₁₁×C₁₀₃ ⋊ C₄
Hall	C₁₁×C₁₀₃ ⋊ Dic₅
Hall	C₅×C₁₀₃ ⋊ Dic₁₁
Sylow	C₁₀₃ ⋊ Dic₅₅
Hall	C₅×C₁₁ ⋊ Dic₁₀₃
Sylow	C₁₁ ⋊ Dic₅₁₅
Sylow	C₅ ⋊ Dic₁₁₃₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₅₆₆₅
commutator, hypocenter	C₅×C₁₁×C₁₀₃	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₁₁×C₁₀₃	C₂
LCS	G ▷ C₅×C₁₁×C₁₀₃
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 22660

A minimal faithful permutation representation — 2 generators on 123 points:

(1,2)(3,4)(5,6,7,8,9)(10,11,12,13,14,15,16,17,18,19,20)(21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3,2,4)(6,9)(7,8)(11,20)(12,19)(13,18)(14,17)(15,16)(22,123)(23,122)(24,121)(25,120)(26,119)(27,118)(28,117)(29,116)(30,115)(31,114)(32,113)(33,112)(34,111)(35,110)(36,109)(37,108)(38,107)(39,106)(40,105)(41,104)(42,103)(43,102)(44,101)(45,100)(46,99)(47,98)(48,97)(49,96)(50,95)(51,94)(52,93)(53,92)(54,91)(55,90)(56,89)(57,88)(58,87)(59,86)(60,85)(61,84)(62,83)(63,82)(64,81)(65,80)(66,79)(67,78)(68,77)(69,76)(70,75)(71,74)(72,73)

Polycyclic presentation:

2,5,11,103,2|a^2=e,b^a=b^4,c^a=c^10,d^a=d^102

Conjugacy classes & character table

5668 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₁₁₃₃₀
maximal normal subgroup of	Dic₁₁₃₃₀

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(22660, 9);

Dic5665 #36383602