#72269413

order	12096
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Hall	C₃²×ΓL(1,𝔽₈) ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₇ ⋊ C₃×#10595100

Subgroups

	subgroup	G/H
center	C₂	C₂×#72222046
commutator, cosocle	C₃²×C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂×D₃²×AGL(1,𝔽₇)
Fitting	C₂×C₃²×C₄×C₇	C₂³×C₃
socle	C₂×C₃²×C₇	C₂²×C₃×D₄
hypercenter	C₂×C₄	D₃²×AGL(1,𝔽₇)
hypocenter	C₃²×C₇	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃²×C₄×C₇ ▷ C₂×C₃²×C₇ ▷ C₃²×C₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄

3236 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

2^4,3,2^2,3^2,7|a^2=f,f^2=g,b^a=bfg,c^a=cg,f^b=fg,h^a=h^2,h^d=h^2,i^a=h^2i,i^d=i^2,j^a=j^6,j^b=j^6,j^e=j^4

Conjugacy classes & character table

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