Dic₁₄₁₆₀ #72290640

order	56640
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₁

Aliases

dicyclic

Dic₁₄₁₆₀

Splittings

Hall	C₃×C₅×C₅₉ ⋊ Q₆₄
Hall	C₅×C₅₉ ⋊ Dic₄₈
Hall	C₃×C₅₉ ⋊ Dic₈₀
Sylow	C₅₉ ⋊ Dic₂₄₀
Hall	C₃×C₅ ⋊ Dic₉₄₄
Sylow	C₅ ⋊ Dic₂₈₃₂
Sylow	C₃ ⋊ Dic₄₇₂₀

Subgroups

	subgroup	G/H
center	C₂	D₁₄₁₆₀
commutator, cosocle	C₃×C₅×C₁₆×C₅₉	C₂²
Frattini	C₁₆	C₂×D₈₈₅
Fitting	C₃×C₅×C₃₂×C₅₉	C₂
socle, prime radical	C₂×C₃×C₅×C₅₉	D₁₆
hypercenter	C₃₂	D₈₈₅
hypocenter	C₃×C₅×C₅₉	Q₆₄
LCS	G ▷ C₃×C₅×C₁₆×C₅₉ ▷ C₃×C₅×C₈×C₅₉ ▷ C₃×C₄×C₅×C₅₉ ▷ C₂×C₃×C₅×C₅₉ ▷ C₃×C₅×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ C₃₂

120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 131
transitive degree	μ_t(G) = 56640

A minimal faithful permutation representation — 2 generators on 131 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)(33,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34)(65,66,67)(68,69,70,71,72)(73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131);(1,33,17,49)(2,34,18,50)(3,35,19,51)(4,36,20,52)(5,37,21,53)(6,38,22,54)(7,39,23,55)(8,40,24,56)(9,41,25,57)(10,42,26,58)(11,43,27,59)(12,44,28,60)(13,45,29,61)(14,46,30,62)(15,47,31,63)(16,48,32,64)(66,67)(69,72)(70,71)(74,131)(75,130)(76,129)(77,128)(78,127)(79,126)(80,125)(81,124)(82,123)(83,122)(84,121)(85,120)(86,119)(87,118)(88,117)(89,116)(90,115)(91,114)(92,113)(93,112)(94,111)(95,110)(96,109)(97,108)(98,107)(99,106)(100,105)(101,104)(102,103)

Polycyclic presentation:

2^6,3,5,59|a^2=f,b^2=cd,c^2=de,d^2=ef,e^2=f,b^a=bc,c^a=cd,d^a=de,e^a=ef,g^a=g^2,h^a=h^4,i^a=i^58

Conjugacy classes & character table

14163 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", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 5, 59]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5*F.6);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 3, 1, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.9^58);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

Dic14160 #72290640