Dic₁₂₅₀₅ #72286792

order	50020
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₂₅₀₅

Splittings

derived, Hall	C₅×C₄₁×C₆₁ ⋊ C₄
Hall	C₄₁×C₆₁ ⋊ Dic₅
Hall	C₅×C₆₁ ⋊ Dic₄₁
Hall	C₅×C₄₁ ⋊ Dic₆₁
Sylow	C₆₁ ⋊ Dic₂₀₅
Sylow	C₄₁ ⋊ Dic₃₀₅
Sylow	C₅ ⋊ Dic₂₅₀₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₂₅₀₅
commutator, hypocenter	C₅×C₄₁×C₆₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₄₁×C₆₁	C₂
LCS	G ▷ C₅×C₄₁×C₆₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 111
transitive degree	μ_t(G) = 50020

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

Polycyclic presentation:

2^2,5,41,61|a^2=b,c^a=c^4,d^a=d^40,e^a=e^60

Conjugacy classes & character table

12508 conjugacy classes. The character table is too large to display.

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e");;
gap> coll := SingleCollector(F, [2, 2, 5, 41, 61]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^40);;
gap> SetConjugate(coll, 5, 1, F.5^60);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic12505 #72286792