Dic₄₀₀₅ #30630803

order	16020
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₀₀₅
unambiguous semidirect product	C₉⋊Dic₄₄₅

Splittings

derived, Hall	C₅×C₉×C₈₉ ⋊ C₄
Hall	C₉×C₈₉ ⋊ Dic₅
Hall	C₅×C₈₉ ⋊ Dic₉
Sylow	C₈₉ ⋊ Dic₄₅
Hall	C₅×C₉ ⋊ Dic₈₉
Sylow	C₉ ⋊ Dic₄₄₅
Sylow	C₅ ⋊ Dic₈₀₁

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₀₀₅
commutator, hypocenter	C₅×C₉×C₈₉	C₄
Frattini	C₂×C₃	D₁₃₃₅
Fitting, cosocle	C₂×C₅×C₉×C₈₉	C₂
socle, prime radical	C₂×C₃×C₅×C₈₉	D₃
LCS	G ▷ C₅×C₉×C₈₉
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 107
transitive degree	μ_t(G) = 16020

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

Polycyclic presentation:

2,3,5,89,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^88,f^a=f^2

Conjugacy classes & character table

4008 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₈₀₁₀, Dic₁₂₀₁₅
maximal normal subgroup of	Dic₈₀₁₀
maximal quotient of	Dic₁₂₀₁₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(16020, 8);

Dic4005 #30630803