#72277632

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂⁶×C₃ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#3484 ⋊ C₂×C₃
Hall	#14262412 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #62817
Sylow	C₇ ⋊ C₃×#62817

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁵×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁶×C₃
Fitting	C₇×#3484	C₂×C₃
hypercenter	#3484	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#62817
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #3484

18988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 112

Polycyclic presentation:

2^6,3,2^2,7|b^a=bh,c^b=ch,d^c=di,e^b=ei,f^c=fh,j^a=j^6,j^g=j^2

Conjugacy classes & character table

272 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, 2, 2, 3, 2, 2, 7]);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.8);;
gap> SetConjugate(coll, 10, 1, F.10^6);;
gap> SetConjugate(coll, 10, 7, F.10^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);