#72247000

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃²×C₇×D₄ ⋊ C₂²×C₃
Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×#41
Sylow	C₇ ⋊ C₃×#64312

Subgroups

	subgroup	G/H
center, Frattini	C₂	C₂²×D₃×C₃⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂×C₃²×C₇	C₂⁴×C₃
Fitting	C₃²×C₇×D₄	C₂²×C₃
hypercenter	D₄	D₃×C₃⋊AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂×C₃×#41
LCS	G ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ D₄

1612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

180 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×#64312
maximal normal subgroup of	AGL(1,𝔽₇)×#64312

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