#72245201

order	14112
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃²×C₄ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Hall	#13096284 ⋊ C₃²
Hall	ΓL(1,𝔽₈)² ⋊ C₂×M₁₆
Sylow	C₇² ⋊ C₂×C₃²×M₁₆

Subgroups

	subgroup	G/H
center, Frattini	C₄	C₂×AGL(1,𝔽₇)²
commutator, socle	C₂×C₇²	C₂²×C₃²×C₄
Fitting	C₂×C₄×C₇²	C₂²×C₃²
prime radical	C₂²×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)	C₄
cosocle	C₄×C₇²	C₂³×C₃²
hypercenter	C₂×C₄	AGL(1,𝔽₇)²
hypocenter	C₇²	C₂×C₃²×M₁₆
LCS	G ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₄ ◁ C₂×C₄

870 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

248 conjugacy classes — view the full character table.

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> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^6);;
gap> SetConjugate(coll, 8, 3, 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.8^6*F.9);;
gap> SetConjugate(coll, 9, 3, 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);