D₂₈₆₂₀ #72292241

order	57240
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₆₂₀

Splittings

Fitting	C₄×C₅×C₂₇×C₅₃ ⋊ C₂
Hall	C₅×C₂₇×C₅₃ ⋊ D₄
Hall	C₂₇×C₅₃ ⋊ D₂₀
Hall	C₅×C₅₃ ⋊ D₁₀₈
Hall	C₅×C₂₇ ⋊ D₂₁₂
Sylow	C₅₃ ⋊ D₅₄₀
Sylow	C₂₇ ⋊ D₁₀₆₀
Sylow	C₅ ⋊ D₅₇₂₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₅₅
commutator, cosocle	C₂×C₅×C₂₇×C₅₃	C₂²
Frattini	C₂×C₉	C₂×D₇₉₅
Fitting	C₄×C₅×C₂₇×C₅₃	C₂
socle	C₂×C₃×C₅×C₅₃	C₂×D₉
hypercenter	C₄	D₇₁₅₅
hypocenter	C₅×C₂₇×C₅₃	D₄
LCS	G ▷ C₂×C₅×C₂₇×C₅₃ ▷ C₅×C₂₇×C₅₃
UCS	C₁ ◁ C₂ ◁ C₄

128 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 89
transitive degree	μ_t(G) = 28620

A minimal faithful permutation representation — 2 generators on 89 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);(2,4)(6,31)(7,30)(8,29)(9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,22)(16,21)(17,20)(18,19)(33,36)(34,35)(38,89)(39,88)(40,87)(41,86)(42,85)(43,84)(44,83)(45,82)(46,81)(47,80)(48,79)(49,78)(50,77)(51,76)(52,75)(53,74)(54,73)(55,72)(56,71)(57,70)(58,69)(59,68)(60,67)(61,66)(62,65)(63,64)

Polycyclic presentation:

2^3,3,5,53,3^2|b^2=c,d^3=g,g^3=h,b^a=bc,d^a=d^2g^2h^2,e^a=e^4,f^a=f^52,g^a=g^2h^2,h^a=h^2

Conjugacy classes & character table

14313 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, 3, 5, 53, 3, 3]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 7, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2*F.7^2*F.8^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^52);;
gap> SetConjugate(coll, 7, 1, F.7^2*F.8^2);;
gap> SetConjugate(coll, 8, 1, F.8^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28620 #72292241