#72218565

order	2688
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting	C₇×#440 ⋊ C₂×C₃
Hall	#11737218 ⋊ C₃
Hall	ΓL(1,𝔽₈) ⋊ C₄×AGL(1,ℤ/8)
Sylow	C₇ ⋊ C₃×C₄×AGL(1,ℤ/8)

Subgroups

	subgroup	G/H
center	C₂²	C₂×D₄×AGL(1,𝔽₇)
commutator	C₄×C₇	C₂³×C₃×C₄
Frattini	C₂×C₄	C₂³×AGL(1,𝔽₇)
Fitting	C₇×#440	C₂×C₃
socle	C₂²×C₇	C₂²×C₃×D₄
prime radical	C₂×#6477⋊C₃	C₂
cosocle	C₂×C₄×C₇	C₂⁴×C₃
hypercenter	#440	AGL(1,𝔽₇)
hypocenter	C₇	C₃×C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₇ ▷ C₂×C₇ ▷ C₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

1096 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72218566, C₄×AGL(1,ℤ/8)×AGL(1,𝔽₇)
maximal normal subgroup of	C₄×AGL(1,ℤ/8)×AGL(1,𝔽₇)
maximal quotient of	#72218566

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