Dic₄₅₄₅ #32911376

order	18180
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic	Dic₄₅₄₅
unambiguous semidirect product	C₉⋊Dic₅₀₅

Splittings

derived, Hall	C₅×C₉×C₁₀₁ ⋊ C₄
Hall	C₉×C₁₀₁ ⋊ Dic₅
Hall	C₅×C₁₀₁ ⋊ Dic₉
Sylow	C₁₀₁ ⋊ Dic₄₅
Hall	C₅×C₉ ⋊ Dic₁₀₁
Sylow	C₉ ⋊ Dic₅₀₅
Sylow	C₅ ⋊ Dic₉₀₉

Subgroups

	subgroup	G/H
center, hypercenter	C₂	D₄₅₄₅
commutator, hypocenter	C₅×C₉×C₁₀₁	C₄
Frattini	C₂×C₃	D₁₅₁₅
Fitting, cosocle	C₂×C₅×C₉×C₁₀₁	C₂
socle, prime radical	C₂×C₃×C₅×C₁₀₁	D₃
LCS	G ▷ C₅×C₉×C₁₀₁
UCS	C₁ ◁ C₂

36 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 119
transitive degree	μ_t(G) = 18180

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);(1,3,2,4)(6,13)(7,12)(8,11)(9,10)(15,18)(16,17)(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:

2,3,5,101,2,3|a^2=e,b^3=f^2,b^a=b^2f,c^a=c^4,d^a=d^100,f^a=f^2

Conjugacy classes & character table

4548 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	Dic₉₀₉₀, Dic₁₃₆₃₅
maximal normal subgroup of	Dic₉₀₉₀
maximal quotient of	Dic₁₃₆₃₅

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(18180, 9);

Dic4545 #32911376