Dic₁₃₉₈₁ #72290353

order	55924
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₉₈₁

Splittings

derived, Hall	C₁₁×C₃₁×C₄₁ ⋊ C₄
Hall	C₃₁×C₄₁ ⋊ Dic₁₁
Hall	C₁₁×C₄₁ ⋊ Dic₃₁
Hall	C₁₁×C₃₁ ⋊ Dic₄₁
Sylow	C₄₁ ⋊ Dic₃₄₁
Sylow	C₃₁ ⋊ Dic₄₅₁
Sylow	C₁₁ ⋊ Dic₁₂₇₁

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₉₈₁
commutator, hypocenter	C₁₁×C₃₁×C₄₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₃₁×C₄₁	C₂
LCS	G ▷ C₁₁×C₃₁×C₄₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 55924

A minimal faithful permutation representation — 2 generators on 87 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,46)(18,45)(19,44)(20,43)(21,42)(22,41)(23,40)(24,39)(25,38)(26,37)(27,36)(28,35)(29,34)(30,33)(31,32)(48,87)(49,86)(50,85)(51,84)(52,83)(53,82)(54,81)(55,80)(56,79)(57,78)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72)(64,71)(65,70)(66,69)(67,68)

Polycyclic presentation:

2^2,11,31,41|a^2=b,c^a=c^10,d^a=d^30,e^a=e^40

Conjugacy classes & character table

13984 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, 11, 31, 41]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^40);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13981 #72290353