#72254381

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Hall	C₃×ΓL(1,𝔽₈) ⋊ #453
Sylow	C₇ ⋊ C₃×#5241

Subgroups

	subgroup	G/H
center	C₂	#72244176
commutator, cosocle	C₂×C₃×C₄×C₇	C₂³×C₃
Frattini	C₂×C₄	C₂×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×#41	C₂²×C₃
socle	C₂×C₃×C₇	C₃×#120
hypercenter	#41	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₃×#453
LCS	G ▷ C₂×C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^3,3,2^3,3,7|f^2=g,b^a=bf,c^a=cg,c^b=ce,e^a=eg,f^a=fg,f^b=fg,f^c=fg,h^b=h^2,i^c=i^6,i^d=i^4

Conjugacy classes & character table

120 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#5241, D₃×#89970⋊C₃, #72269426
maximal normal subgroup of	AGL(1,𝔽₇)×#5241, D₃×#89970⋊C₃, #72269426

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