D₂₂₈₉₆ #72278313

order	45792
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₂₈₉₆

Splittings

Fitting	C₁₆×C₂₇×C₅₃ ⋊ C₂
Hall	C₂₇×C₅₃ ⋊ D₁₆
Sylow	C₅₃ ⋊ D₄₃₂
Sylow	C₂₇ ⋊ D₈₄₈

Subgroups

	subgroup	G/H
center	C₂	D₁₁₄₄₈
commutator, cosocle	C₈×C₂₇×C₅₃	C₂²
Frattini	C₈×C₉	C₂×D₁₅₉
Fitting	C₁₆×C₂₇×C₅₃	C₂
socle	C₂×C₃×C₅₃	D₇₂
hypercenter	C₁₆	D₁₄₃₁
hypocenter	C₂₇×C₅₃	D₁₆
LCS	G ▷ C₈×C₂₇×C₅₃ ▷ C₄×C₂₇×C₅₃ ▷ C₂×C₂₇×C₅₃ ▷ C₂₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆

112 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 96
transitive degree	μ_t(G) = 22896

A minimal faithful permutation representation — 2 generators on 96 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);(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)(45,96)(46,95)(47,94)(48,93)(49,92)(50,91)(51,90)(52,89)(53,88)(54,87)(55,86)(56,85)(57,84)(58,83)(59,82)(60,81)(61,80)(62,79)(63,78)(64,77)(65,76)(66,75)(67,74)(68,73)(69,72)(70,71)

Polycyclic presentation:

2^5,3,53,3^2|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^52,h^a=h^2i^2,i^a=i^2

Conjugacy classes & character table

11451 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, 53, 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.7^52);;
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);

D22896 #72278313