D₉₉₄₀ #72215038

order	19880
automorphism	Aut(G) order 33398400
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

dihedral

D₉₉₄₀

Splittings

Fitting	C₄×C₅×C₇×C₇₁ ⋊ C₂
Hall	C₅×C₇×C₇₁ ⋊ D₄
Hall	C₇×C₇₁ ⋊ D₂₀
Hall	C₅×C₇₁ ⋊ D₂₈
Sylow	C₇₁ ⋊ D₁₄₀
Hall	C₅×C₇ ⋊ D₂₈₄
Sylow	C₇ ⋊ D₁₄₂₀
Sylow	C₅ ⋊ D₁₉₈₈

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂×D₂₄₈₅
commutator, socle, cosocle	C₂×C₅×C₇×C₇₁	C₂²
Fitting	C₄×C₅×C₇×C₇₁	C₂
hypercenter	C₄	D₂₄₈₅
hypocenter	C₅×C₇×C₇₁	D₄
LCS	G ▷ C₂×C₅×C₇×C₇₁ ▷ C₅×C₇×C₇₁
UCS	C₁ ◁ C₂ ◁ C₄

64 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 87
transitive degree	μ_t(G) = 9940

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)(11,16)(12,15)(13,14)(18,87)(19,86)(20,85)(21,84)(22,83)(23,82)(24,81)(25,80)(26,79)(27,78)(28,77)(29,76)(30,75)(31,74)(32,73)(33,72)(34,71)(35,70)(36,69)(37,68)(38,67)(39,66)(40,65)(41,64)(42,63)(43,62)(44,61)(45,60)(46,59)(47,58)(48,57)(49,56)(50,55)(51,54)(52,53)

Polycyclic presentation:

2^3,5,7,71|b^2=c,b^a=bc,d^a=d^4,e^a=e^6,f^a=f^70

Conjugacy classes & character table

4973 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	D₁₉₈₈₀, D₂₉₈₂₀
maximal normal subgroup of	D₁₉₈₈₀
maximal quotient of	Dic₉₉₄₀, D₁₉₈₈₀, D₂₉₈₂₀

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, 5, 7, 71]);;
gap> SetPower(coll, 2, F.3);;
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);;
gap> SetConjugate(coll, 6, 1, F.6^70);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

D9940 #72215038