#72245671

order	4032
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₃×C₇×SD₁₆ ⋊ C₂²×C₃
Hall	C₃×ΓL(1,𝔽₈) ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₇ ⋊ C₃×D₃×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	D₃×D₄×AGL(1,𝔽₇)
commutator, cosocle	C₃×C₄×C₇	C₂⁴×C₃
Frattini	C₄	C₂²×D₃×AGL(1,𝔽₇)
Fitting	C₃×C₇×SD₁₆	C₂²×C₃
socle	C₂×C₃×C₇	C₂²×C₃×D₄
hypercenter	SD₁₆	D₃×AGL(1,𝔽₇)
hypocenter	C₃×C₇	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₃×C₄×C₇ ▷ C₂×C₃×C₇ ▷ C₃×C₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ SD₁₆

1260 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 168

Polycyclic presentation:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₃×AGL(1,ℤ/8)×AGL(1,𝔽₇)
maximal normal subgroup of	D₃×AGL(1,ℤ/8)×AGL(1,𝔽₇)

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