D₂₈₄₂₀ #72291960

order	56840
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₂₈₄₂₀

Splittings

Fitting	C₄×C₅×C₂₉×C₄₉ ⋊ C₂
Hall	C₅×C₂₉×C₄₉ ⋊ D₄
Hall	C₂₉×C₄₉ ⋊ D₂₀
Hall	C₅×C₄₉ ⋊ D₁₁₆
Hall	C₅×C₂₉ ⋊ D₁₉₆
Sylow	C₄₉ ⋊ D₅₈₀
Sylow	C₂₉ ⋊ D₉₈₀
Sylow	C₅ ⋊ D₅₆₈₄

Subgroups

	subgroup	G/H
center	C₂	C₂×D₇₁₀₅
commutator, cosocle	C₂×C₅×C₂₉×C₄₉	C₂²
Frattini	C₂×C₇	C₂×D₁₀₁₅
Fitting	C₄×C₅×C₂₉×C₄₉	C₂
socle	C₂×C₅×C₇×C₂₉	C₂×D₇
hypercenter	C₄	D₇₁₀₅
hypocenter	C₅×C₂₉×C₄₉	D₄
LCS	G ▷ C₂×C₅×C₂₉×C₄₉ ▷ C₅×C₂₉×C₄₉
UCS	C₁ ◁ C₂ ◁ C₄

96 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 28420

A minimal faithful permutation representation — 2 generators on 87 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);(2,4)(6,9)(7,8)(11,58)(12,57)(13,56)(14,55)(15,54)(16,53)(17,52)(18,51)(19,50)(20,49)(21,48)(22,47)(23,46)(24,45)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,38)(32,37)(33,36)(34,35)(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:

2^3,5,7,29,7|b^2=c,e^7=g,b^a=bc,d^a=d^4,e^a=e^6g^6,f^a=f^28,g^a=g^6

Conjugacy classes & character table

14213 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, 5, 7, 29, 7]);;
gap> SetPower(coll, 2, F.3);;
gap> SetPower(coll, 5, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^4);;
gap> SetConjugate(coll, 5, 1, F.5^6*F.7^6);;
gap> SetConjugate(coll, 6, 1, F.6^28);;
gap> SetConjugate(coll, 7, 1, F.7^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D28420 #72291960