D₂₅₄₈₈ #72288643

order	50976
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₅₄₈₈

Splittings

Fitting	C₁₆×C₂₇×C₅₉ ⋊ C₂
Hall	C₂₇×C₅₉ ⋊ D₁₆
Sylow	C₅₉ ⋊ D₄₃₂
Sylow	C₂₇ ⋊ D₉₄₄

Subgroups

	subgroup	G/H
center	C₂	D₁₂₇₄₄
commutator, cosocle	C₈×C₂₇×C₅₉	C₂²
Frattini	C₈×C₉	C₂×D₁₇₇
Fitting	C₁₆×C₂₇×C₅₉	C₂
socle	C₂×C₃×C₅₉	D₇₂
hypercenter	C₁₆	D₁₅₉₃
hypocenter	C₂₇×C₅₉	D₁₆
LCS	G ▷ C₈×C₂₇×C₅₉ ▷ C₄×C₂₇×C₅₉ ▷ C₂×C₂₇×C₅₉ ▷ C₂₇×C₅₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 102
transitive degree	μ_t(G) = 25488

A minimal faithful permutation representation — 2 generators on 102 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);(2,16)(3,15)(4,14)(5,13)(6,12)(7,11)(8,10)(18,43)(19,42)(20,41)(21,40)(22,39)(23,38)(24,37)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(45,102)(46,101)(47,100)(48,99)(49,98)(50,97)(51,96)(52,95)(53,94)(54,93)(55,92)(56,91)(57,90)(58,89)(59,88)(60,87)(61,86)(62,85)(63,84)(64,83)(65,82)(66,81)(67,80)(68,79)(69,78)(70,77)(71,76)(72,75)(73,74)

Polycyclic presentation:

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

Conjugacy classes & character table

12747 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, 3, 59, 3, 3]);;
gap> SetPower(coll, 2, F.3*F.4);;
gap> SetPower(coll, 3, F.4*F.5);;
gap> SetPower(coll, 4, F.5);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
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*F.8^2*F.9^2);;
gap> SetConjugate(coll, 7, 1, F.7^58);;
gap> SetConjugate(coll, 8, 1, F.8^2*F.9^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D25488 #72288643