#72222243

order	6048
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂⁴×C₇ ⋊ C₃×C₃⋊D₃
derived	C₇×A₄² ⋊ C₂×C₃
Sylow	C₇ ⋊ C₃×A₄⋊S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇×A₄²	C₂×C₃
Fitting, socle	C₂⁴×C₇	C₃×C₃⋊D₃
LCS	G ▷ C₇×A₄²
UCS	C₁

456 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2,3^3,7,2^4|c^a=c^2,d^a=d^2,e^a=e^6,e^b=e^2,f^c=g,f^d=g,g^a=fg,g^c=fg,g^d=fg,h^a=ghi,h^c=i,h^d=fgh,i^a=fi,i^c=hi,i^d=fi

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	AGL(1,𝔽₇)×A₄⋊S₄, S₄×A₄⋊AGL(1,𝔽₇), C₂×#72222243
maximal normal subgroup of	AGL(1,𝔽₇)×A₄⋊S₄, S₄×A₄⋊AGL(1,𝔽₇), C₂×#72222243
maximal quotient of	C₂×#72222243, #72246260

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