#72259588

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₃ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#550 ⋊ C₂×C₃
Hall	#11743838 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #3498
Sylow	C₇ ⋊ C₃×#3498

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇	C₂⁵×C₃
Fitting	C₇×#550	C₂×C₃
hypercenter	#550	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#3498
LCS	G ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ #550

3496 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 224

Polycyclic presentation:

2^5,3,2^2,7|d^2=h,e^2=h,b^a=bg,d^a=dg,d^c=dh,e^a=eh,e^b=eh,i^c=i^6,i^f=i^4

Conjugacy classes & character table

157 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#3498
maximal normal subgroup of	AGL(1,𝔽₇)×#3498

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