#72269412

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃×C₇² ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₇² ⋊ C₂×#949
Sylow	C₃ ⋊ #72214529

Subgroups

	subgroup	G/H
center	C₂	C₂×#72222125
commutator, cosocle	C₃×C₄×C₇²	C₂⁴
Frattini	C₄	C₂×D₃×D₇²
Fitting	C₂×C₃×C₄×C₇²	C₂³
socle	C₂×C₃×C₇²	C₂²×D₄
hypercenter	C₂×C₄	D₃×D₇²
hypocenter	C₃×C₇²	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₇² ▷ C₂×C₃×C₇² ▷ C₃×C₇²
UCS	C₁ ◁ C₂ ◁ C₂×C₄

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^6,3,7^2|a^2=e,e^2=f,b^a=bef,c^a=cf,e^b=ef,g^a=g^2,g^b=g^2,h^a=h^6,h^d=h^6,i^a=h^2i,i^d=i^6

Conjugacy classes & character table

330 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");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 3, 7, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 5, F.6);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 4, F.8^6);;
gap> SetConjugate(coll, 9, 1, F.8^2*F.9);;
gap> SetConjugate(coll, 9, 4, F.9^6);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);