#72246104

order	14112
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃² · Schur multiplier (H₂) = C₂⁴×C₃

Aliases

No aliases for this group.

Splittings

Fitting	C₇²×C₂²⋊C₄ ⋊ C₂×C₃²
Hall	#13096523 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ #120
Sylow	C₇² ⋊ C₃²×#120

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
commutator, socle, cosocle	C₂²×C₇²	C₂³×C₃²
Fitting	C₇²×C₂²⋊C₄	C₂×C₃²
hypercenter	C₂²⋊C₄	ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
hypocenter	C₇²	C₃²×#120
LCS	G ▷ C₂²×C₇² ▷ C₇²
UCS	C₁ ◁ C₂² ◁ C₂²⋊C₄

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^3,3^2,2^2,7^2|b^a=bf,c^a=cg,h^a=h^6,h^b=h^6,h^d=h^4,h^e=h^2,i^a=i^6,i^b=i^6,i^d=h^5i,i^e=i^2

Conjugacy classes & character table

206 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#120×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72254299
maximal normal subgroup of	#120×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72254299
maximal quotient of	#72246405, #72246407, #72254299

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