Dic₁₅₂₈₃ #72293094

order	61132
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₈₃

Splittings

derived, Hall	C₁₇×C₂₉×C₃₁ ⋊ C₄
Hall	C₂₉×C₃₁ ⋊ Dic₁₇
Hall	C₁₇×C₃₁ ⋊ Dic₂₉
Hall	C₁₇×C₂₉ ⋊ Dic₃₁
Sylow	C₃₁ ⋊ Dic₄₉₃
Sylow	C₂₉ ⋊ Dic₅₂₇
Sylow	C₁₇ ⋊ Dic₈₉₉

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₂₈₃
commutator, hypocenter	C₁₇×C₂₉×C₃₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₁₇×C₂₉×C₃₁	C₂
LCS	G ▷ C₁₇×C₂₉×C₃₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 81
transitive degree	μ_t(G) = 61132

A minimal faithful permutation representation — 2 generators on 81 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);(1,3,2,4)(6,21)(7,20)(8,19)(9,18)(10,17)(11,16)(12,15)(13,14)(23,50)(24,49)(25,48)(26,47)(27,46)(28,45)(29,44)(30,43)(31,42)(32,41)(33,40)(34,39)(35,38)(36,37)(52,81)(53,80)(54,79)(55,78)(56,77)(57,76)(58,75)(59,74)(60,73)(61,72)(62,71)(63,70)(64,69)(65,68)(66,67)

Polycyclic presentation:

2^2,17,29,31|a^2=b,c^a=c^16,d^a=d^28,e^a=e^30

Conjugacy classes & character table

15286 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, 17, 29, 31]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^16);;
gap> SetConjugate(coll, 4, 1, F.4^28);;
gap> SetConjugate(coll, 5, 1, F.5^30);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15283 #72293094