Dic₁₄₉₂₇ #72292278

order	59708
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₉₂₇

Splittings

derived, Hall	C₁₁×C₂₃×C₅₉ ⋊ C₄
Hall	C₂₃×C₅₉ ⋊ Dic₁₁
Hall	C₁₁×C₅₉ ⋊ Dic₂₃
Hall	C₁₁×C₂₃ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₂₅₃
Sylow	C₂₃ ⋊ Dic₆₄₉
Sylow	C₁₁ ⋊ Dic₁₃₅₇

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₄₉₂₇
commutator, hypocenter	C₁₁×C₂₃×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₁×C₂₃×C₅₉	C₂
LCS	G ▷ C₁₁×C₂₃×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 97
transitive degree	μ_t(G) = 59708

A minimal faithful permutation representation — 2 generators on 97 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);(1,3,2,4)(6,15)(7,14)(8,13)(9,12)(10,11)(17,38)(18,37)(19,36)(20,35)(21,34)(22,33)(23,32)(24,31)(25,30)(26,29)(27,28)(40,97)(41,96)(42,95)(43,94)(44,93)(45,92)(46,91)(47,90)(48,89)(49,88)(50,87)(51,86)(52,85)(53,84)(54,83)(55,82)(56,81)(57,80)(58,79)(59,78)(60,77)(61,76)(62,75)(63,74)(64,73)(65,72)(66,71)(67,70)(68,69)

Polycyclic presentation:

2^2,11,23,59|a^2=b,c^a=c^10,d^a=d^22,e^a=e^58

Conjugacy classes & character table

14930 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, 23, 59]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^10);;
gap> SetConjugate(coll, 4, 1, F.4^22);;
gap> SetConjugate(coll, 5, 1, F.5^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14927 #72292278