D₂₄₂₄₀ #72287005

order	48480
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₄₂₄₀

Splittings

Fitting	C₃×C₅×C₁₆×C₁₀₁ ⋊ C₂
Hall	C₃×C₅×C₁₀₁ ⋊ D₁₆
Hall	C₅×C₁₀₁ ⋊ D₄₈
Hall	C₃×C₁₀₁ ⋊ D₈₀
Sylow	C₁₀₁ ⋊ D₂₄₀
Hall	C₃×C₅ ⋊ D₁₆₁₆
Sylow	C₅ ⋊ D₄₈₄₈
Sylow	C₃ ⋊ D₈₀₈₀

Subgroups

	subgroup	G/H
center	C₂	D₁₂₁₂₀
commutator, cosocle	C₃×C₅×C₈×C₁₀₁	C₂²
Frattini	C₈	C₂×D₁₅₁₅
Fitting	C₃×C₅×C₁₆×C₁₀₁	C₂
socle	C₂×C₃×C₅×C₁₀₁	D₈
hypercenter	C₁₆	D₁₅₁₅
hypocenter	C₃×C₅×C₁₀₁	D₁₆
LCS	G ▷ C₃×C₅×C₈×C₁₀₁ ▷ C₃×C₄×C₅×C₁₀₁ ▷ C₂×C₃×C₅×C₁₀₁ ▷ C₃×C₅×C₁₀₁
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 125
transitive degree	μ_t(G) = 24240

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

Polycyclic presentation:

2^5,3,5,101|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^2,g^a=g^4,h^a=h^100

Conjugacy classes & character table

12123 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 3, 5, 101]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
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, 6, 1, F.6^2);;
gap> SetConjugate(coll, 7, 1, F.7^4);;
gap> SetConjugate(coll, 8, 1, F.8^100);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D24240 #72287005