Dic₁₄₉₁₁ #72292255

order	59644
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₁₁

Splittings

derived, Hall	C₁₃×C₃₁×C₃₇ ⋊ C₄
Hall	C₃₁×C₃₇ ⋊ Dic₁₃
Hall	C₁₃×C₃₇ ⋊ Dic₃₁
Hall	C₁₃×C₃₁ ⋊ Dic₃₇
Sylow	C₃₇ ⋊ Dic₄₀₃
Sylow	C₃₁ ⋊ Dic₄₈₁
Sylow	C₁₃ ⋊ Dic₁₁₄₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₉₁₁
commutator, hypocenter	C₁₃×C₃₁×C₃₇	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₃×C₃₁×C₃₇	C₂
LCS	G ▷ C₁₃×C₃₁×C₃₇
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 59644

A minimal faithful permutation representation — 2 generators on 85 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);(1,3,2,4)(6,17)(7,16)(8,15)(9,14)(10,13)(11,12)(19,48)(20,47)(21,46)(22,45)(23,44)(24,43)(25,42)(26,41)(27,40)(28,39)(29,38)(30,37)(31,36)(32,35)(33,34)(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:

2^2,13,31,37|a^2=b,c^a=c^12,d^a=d^30,e^a=e^36

Conjugacy classes & character table

14914 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, 13, 31, 37]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^12);;
gap> SetConjugate(coll, 4, 1, F.4^30);;
gap> SetConjugate(coll, 5, 1, F.5^36);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14911 #72292255