Dic₁₃₈₆₅ #72290165

order	55460
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₃₈₆₅

Splittings

derived, Hall	C₅×C₄₇×C₅₉ ⋊ C₄
Hall	C₄₇×C₅₉ ⋊ Dic₅
Hall	C₅×C₅₉ ⋊ Dic₄₇
Hall	C₅×C₄₇ ⋊ Dic₅₉
Sylow	C₅₉ ⋊ Dic₂₃₅
Sylow	C₄₇ ⋊ Dic₂₉₅
Sylow	C₅ ⋊ Dic₂₇₇₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₃₈₆₅
commutator, hypocenter	C₅×C₄₇×C₅₉	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₄₇×C₅₉	C₂
LCS	G ▷ C₅×C₄₇×C₅₉
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 115
transitive degree	μ_t(G) = 55460

A minimal faithful permutation representation — 2 generators on 115 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)(57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115);(1,3,2,4)(6,9)(7,8)(11,56)(12,55)(13,54)(14,53)(15,52)(16,51)(17,50)(18,49)(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)(58,115)(59,114)(60,113)(61,112)(62,111)(63,110)(64,109)(65,108)(66,107)(67,106)(68,105)(69,104)(70,103)(71,102)(72,101)(73,100)(74,99)(75,98)(76,97)(77,96)(78,95)(79,94)(80,93)(81,92)(82,91)(83,90)(84,89)(85,88)(86,87)

Polycyclic presentation:

2^2,5,47,59|a^2=b,c^a=c^4,d^a=d^46,e^a=e^58

Conjugacy classes & character table

13868 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, 47, 59]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^46);;
gap> SetConjugate(coll, 5, 1, F.5^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic13865 #72290165