#72246128

order	18816
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×#120 ⋊ C₂²×C₃
Hall	#72215173 ⋊ C₃
Hall	C₇×ΓL(1,𝔽₈) ⋊ C₂×B(4,𝔽₂)
Sylow	C₇² ⋊ C₂×C₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72245959
commutator, cosocle	C₂³×C₇²	C₂⁴×C₃
Frattini	C₂³	C₂²×D₇×AGL(1,𝔽₇)
Fitting	C₇²×#120	C₂²×C₃
socle	C₂×C₇²	C₂×C₃×#120
hypercenter	#120	D₇×AGL(1,𝔽₇)
hypocenter	C₇²	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #120

4403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^4,3,2^3,7^2|a^2=h,c^a=cg,c^b=cf,f^a=fh,g^b=gh,i^a=i^6,i^d=i^6,i^e=i^4,j^a=i^6j,j^d=j^6,j^e=i^5j

Conjugacy classes & character table

298 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, 7, 7]);;
gap> SetPower(coll, 1, F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 7, 2, F.7*F.8);;
gap> SetConjugate(coll, 9, 1, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^6);;
gap> SetConjugate(coll, 9, 5, F.9^4);;
gap> SetConjugate(coll, 10, 1, F.9^6*F.10);;
gap> SetConjugate(coll, 10, 4, F.10^6);;
gap> SetConjugate(coll, 10, 5, F.9^5*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);