#72259697

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#1715 ⋊ C₂×C₃
Hall	#13343040 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #19745
Sylow	C₇ ⋊ C₃×#19745

Subgroups

	subgroup	G/H
center	C₂²	#127×AGL(1,𝔽₇)
commutator, cosocle	C₄²×C₇	C₂⁴×C₃
Frattini	C₄²	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#1715	C₂×C₃
socle	C₂²×C₇	C₂×C₃×#127
hypercenter	#1715	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#19745
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1715

1988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 448

Polycyclic presentation:

2^4,3,2^4,7|d^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cf,c^b=ch,d^a=dg,d^b=dgi,d^c=dg,f^a=fh,f^b=fh,f^c=fh,g^a=gi,g^b=gi,g^c=gi,j^a=j^6,j^b=j^6,j^e=j^4

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#19745
maximal normal subgroup of	AGL(1,𝔽₇)×#19745

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