Dic₁₅₂₆₅ #72292984

order	61060
automorphism	?
homology	Abelianization (H₁) = C₄ · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₅₂₆₅

Splittings

derived, Hall	C₅×C₄₃×C₇₁ ⋊ C₄
Hall	C₄₃×C₇₁ ⋊ Dic₅
Hall	C₅×C₇₁ ⋊ Dic₄₃
Hall	C₅×C₄₃ ⋊ Dic₇₁
Sylow	C₇₁ ⋊ Dic₂₁₅
Sylow	C₄₃ ⋊ Dic₃₅₅
Sylow	C₅ ⋊ Dic₃₀₅₃

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	D₁₅₂₆₅
commutator, hypocenter	C₅×C₄₃×C₇₁	C₄
Fitting, socle, prime radical, cosocle	C₂×C₅×C₄₃×C₇₁	C₂
LCS	G ▷ C₅×C₄₃×C₇₁
UCS	C₁ ◁ C₂

24 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 123
transitive degree	μ_t(G) = 61060

A minimal faithful permutation representation — 2 generators on 123 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)(53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123);(1,3,2,4)(6,9)(7,8)(11,52)(12,51)(13,50)(14,49)(15,48)(16,47)(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)(54,123)(55,122)(56,121)(57,120)(58,119)(59,118)(60,117)(61,116)(62,115)(63,114)(64,113)(65,112)(66,111)(67,110)(68,109)(69,108)(70,107)(71,106)(72,105)(73,104)(74,103)(75,102)(76,101)(77,100)(78,99)(79,98)(80,97)(81,96)(82,95)(83,94)(84,93)(85,92)(86,91)(87,90)(88,89)

Polycyclic presentation:

2^2,5,43,71|a^2=b,c^a=c^4,d^a=d^42,e^a=e^70

Conjugacy classes & character table

15268 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, 43, 71]);;
gap> SetPower(coll, 1, F.2);;
gap> SetConjugate(coll, 3, 1, F.3^4);;
gap> SetConjugate(coll, 4, 1, F.4^42);;
gap> SetConjugate(coll, 5, 1, F.5^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic15265 #72292984