D₂₈₂₀₈ #72291544

order	56416
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₂₀₈

Splittings

Fitting	C₁₆×C₄₁×C₄₃ ⋊ C₂
Hall	C₄₁×C₄₃ ⋊ D₁₆
Sylow	C₄₃ ⋊ D₆₅₆
Sylow	C₄₁ ⋊ D₆₈₈

Subgroups

	subgroup	G/H
center	C₂	D₁₄₁₀₄
commutator, cosocle	C₈×C₄₁×C₄₃	C₂²
Frattini	C₈	C₂×D₁₇₆₃
Fitting	C₁₆×C₄₁×C₄₃	C₂
socle	C₂×C₄₁×C₄₃	D₈
hypercenter	C₁₆	D₁₇₆₃
hypocenter	C₄₁×C₄₃	D₁₆
LCS	G ▷ C₈×C₄₁×C₄₃ ▷ C₄×C₄₁×C₄₃ ▷ C₂×C₄₁×C₄₃ ▷ C₄₁×C₄₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

56 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 100
transitive degree	μ_t(G) = 28208

A minimal faithful permutation representation — 2 generators on 100 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,57)(19,56)(20,55)(21,54)(22,53)(23,52)(24,51)(25,50)(26,49)(27,48)(28,47)(29,46)(30,45)(31,44)(32,43)(33,42)(34,41)(35,40)(36,39)(37,38)(59,100)(60,99)(61,98)(62,97)(63,96)(64,95)(65,94)(66,93)(67,92)(68,91)(69,90)(70,89)(71,88)(72,87)(73,86)(74,85)(75,84)(76,83)(77,82)(78,81)(79,80)

Polycyclic presentation:

2^5,41,43|b^2=cd,c^2=de,d^2=e,b^a=bc,c^a=cd,d^a=de,f^a=f^40,g^a=g^42

Conjugacy classes & character table

14107 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 41, 43]);;
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.6^40);;
gap> SetConjugate(coll, 7, 1, F.7^42);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28208 #72291544