#72269461

order	8064
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×#99 ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×B(4,𝔽₂)
Sylow	C₇ ⋊ C₃×D₃×B(4,𝔽₂)

Subgroups

	subgroup	G/H
center	C₂	#72244221
commutator, cosocle	C₂³×C₃×C₇	C₂⁴×C₃
Frattini	C₂³	C₂²×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×#99	C₂²×C₃
socle	C₂×C₃×C₇	C₂×C₃×#120
hypercenter	#99	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×B(4,𝔽₂)
LCS	G ▷ C₂³×C₃×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #99

4270 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

171 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,𝔽₇)×B(4,𝔽₂)
maximal normal subgroup of	D₃×AGL(1,𝔽₇)×B(4,𝔽₂)

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