#72259912

order	3584
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#33688 ⋊ C₂
Sylow	C₇ ⋊ #489347

Subgroups

	subgroup	G/H
center	C₂	#13329861
commutator, cosocle	C₄×C₇×C₈	C₂⁴
Frattini	C₄×C₈	C₂³×D₇
Fitting	C₇×#33688	C₂
socle	C₂×C₇	#18524
hypercenter	#33688	D₇
hypocenter	C₇	#489347
LCS	G ▷ C₄×C₇×C₈ ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ #33688

1920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Polycyclic presentation:

2^9,7|a^2=i,e^2=g,f^2=h,h^2=i,b^a=bf,c^a=chi,c^b=ce,d^b=de,d^c=dg,e^a=eh,e^b=eg,e^c=eg,e^d=eg,f^a=fhi,f^b=fhi,f^c=fh,f^d=fhi,g^a=gi,h^a=hi,h^b=hi,h^c=hi,h^d=hi,j^c=j^6,j^d=j^6

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₇×#489347
maximal normal subgroup of	D₇×#489347

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.9);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 8, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.8*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.8*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 10, 3, F.10^6);;
gap> SetConjugate(coll, 10, 4, F.10^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);