D₅₃₄₀ #72211646

order	10680
automorphism	Aut(G) order 7518720
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₅₃₄₀

Splittings

Fitting	C₃×C₄×C₅×C₈₉ ⋊ C₂
Hall	C₃×C₅×C₈₉ ⋊ D₄
Hall	C₅×C₈₉ ⋊ D₁₂
Hall	C₃×C₈₉ ⋊ D₂₀
Sylow	C₈₉ ⋊ D₆₀
Hall	C₃×C₅ ⋊ D₃₅₆
Sylow	C₅ ⋊ D₁₀₆₈
Sylow	C₃ ⋊ D₁₇₈₀

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₁₃₃₅
commutator, socle, cosocle	C₂×C₃×C₅×C₈₉	C₂²
Fitting	C₃×C₄×C₅×C₈₉	C₂
hypercenter	C₄	D₁₃₃₅
hypocenter	C₃×C₅×C₈₉	D₄
LCS	G ▷ C₂×C₃×C₅×C₈₉ ▷ C₃×C₅×C₈₉
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 101
transitive degree	μ_t(G) = 5340

A minimal faithful permutation representation — 2 generators on 101 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);(2,4)(6,7)(9,12)(10,11)(14,101)(15,100)(16,99)(17,98)(18,97)(19,96)(20,95)(21,94)(22,93)(23,92)(24,91)(25,90)(26,89)(27,88)(28,87)(29,86)(30,85)(31,84)(32,83)(33,82)(34,81)(35,80)(36,79)(37,78)(38,77)(39,76)(40,75)(41,74)(42,73)(43,72)(44,71)(45,70)(46,69)(47,68)(48,67)(49,66)(50,65)(51,64)(52,63)(53,62)(54,61)(55,60)(56,59)(57,58)

Polycyclic presentation:

2^3,3,5,89|b^2=c,b^a=bc,d^a=d^2,e^a=e^4,f^a=f^88

Conjugacy classes & character table

2673 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₀₆₈₀, D₁₆₀₂₀, D₂₆₇₀₀
maximal normal subgroup of	D₁₀₆₈₀
maximal quotient of	D₁₀₆₈₀, Dic₅₃₄₀, D₁₆₀₂₀, …

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");;
gap> coll := SingleCollector(F, [2, 2, 2, 3, 5, 89]);;
gap> SetPower(coll, 2, F.3);;
gap> SetConjugate(coll, 2, 1, F.2*F.3);;
gap> SetConjugate(coll, 4, 1, F.4^2);;
gap> SetConjugate(coll, 5, 1, F.5^4);;
gap> SetConjugate(coll, 6, 1, F.6^88);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D5340 #72211646