#72259472

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#492 ⋊ C₂×C₃
Hall	#11729013 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #1715
Sylow	C₇ ⋊ C₃×#1715

Subgroups

	subgroup	G/H
center	C₂²	#6503⋊C₃
commutator, cosocle	C₄²×C₇	C₂³×C₃
Frattini	C₄²	C₂²×AGL(1,𝔽₇)
Fitting	C₇×#492	C₂×C₃
socle	C₂²×C₇	C₃×#127
prime radical	#89845⋊C₃	C₂
hypercenter	#492	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#1715
LCS	G ▷ C₄²×C₇ ▷ C₂²×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

440 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 448

Polycyclic presentation:

2^3,3,2^4,7|a^2=e,c^2=fh,e^2=g,f^2=h,b^a=beg,c^a=ch,c^b=cf,e^b=eg,f^b=fh,i^c=i^6,i^d=i^4

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#1715, #72259679, #72259697
maximal normal subgroup of	AGL(1,𝔽₇)×#1715, #72259679, #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, 3, 2, 2, 2, 2, 7]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.6*F.8);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.5*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 9, 3, F.9^6);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);