#72214550

order	9408
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₂×C₇²×D₄ ⋊ C₂²×C₃
Hall	#72214529 ⋊ C₃
Hall	#3861 ⋊ C₂×AGL(1,ℤ/8)
Sylow	C₇² ⋊ C₂×C₃×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂	C₂×#72210376
commutator, cosocle	C₄×C₇²	C₂⁴×C₃
Frattini	C₄	C₂²×#64423⋊C₂
Fitting	C₂×C₇²×D₄	C₂²×C₃
socle	C₂×C₇²	C₂²×C₃×D₄
hypercenter	C₂×D₄	#64423⋊C₂
hypocenter	C₇²	C₂×C₃×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇² ▷ C₂×C₇² ▷ C₇²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×D₄

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72215081
maximal normal subgroup of	#72215081
holomorph of	C₇⋊C₈

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