#72277927

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#2403 ⋊ C₂×C₃
Hall	#13502566 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #32898
Sylow	C₇ ⋊ C₃×#32898

Subgroups

	subgroup	G/H
center	C₂²	C₂×#6500⋊C₃
commutator, cosocle	C₂⁴×C₇	C₂⁴×C₃
Frattini	C₂⁴	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#2403	C₂×C₃
socle	C₂²×C₇	C₂×C₃×#120
hypercenter	#2403	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#32898
LCS	G ▷ C₂⁴×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ #2403

8720 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 224

Polycyclic presentation:

2^4,3,2^4,7|a^2=i,d^2=f,b^a=bi,d^a=dh,d^b=dg,d^c=df,g^a=gi,h^b=hi,j^b=j^6,j^e=j^4

Conjugacy classes & character table

185 conjugacy classes — view the full character table.

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