#72259378

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#496 ⋊ C₂×C₃
Hall	#11739290 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3151
Sylow	C₇ ⋊ C₃×#3151

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×AGL(1,𝔽₇)
commutator, cosocle	C₂×C₄×C₇	C₂⁴×C₃
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#496	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×D₄
hypercenter	#496	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3151
LCS	G ▷ C₂×C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #496

1312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 224

Polycyclic presentation:

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

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3151
maximal normal subgroup of	AGL(1,𝔽₇)×#3151
maximal quotient of	#72259644, #72259697

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