#72276889

order	5376
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#1438 ⋊ C₂×C₃
Hall	#13301308 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ #15947
Sylow	C₇ ⋊ C₃×#15947

Subgroups

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×AGL(1,𝔽₇)
commutator, socle, cosocle	C₂⁴×C₇	C₂⁴×C₃
Fitting	C₇×#1438	C₂×C₃
hypercenter	#1438	AGL(1,𝔽₇)
hypocenter	C₇	C₃×#15947
LCS	G ▷ C₂⁴×C₇ ▷ C₇
UCS	C₁ ◁ C₂⁴ ◁ #1438

12292 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 448

Polycyclic presentation:

2^4,3,2^4,7|b^2=g,b^a=bi,c^a=ch,c^b=cf,d^a=di,d^b=dg,j^d=j^6,j^e=j^4

Conjugacy classes & character table

218 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#15947
maximal normal subgroup of	AGL(1,𝔽₇)×#15947

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