#72222237

order	2016
automorphism	?
homology	Abelianization (H₁) = C₂ · Schur multiplier (H₂) = C₂²×C₃

Aliases

No aliases for this group.

Splittings

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

Subgroups

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

190 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 23 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:

2,3^2,7,2^4|b^a=b^2,c^a=c^2,d^a=d^6,e^b=f,e^c=f,f^a=ef,f^b=ef,f^c=ef,g^a=fgh,g^b=h,g^c=efg,h^a=eh,h^b=gh,h^c=eh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222243, D₇×A₄⋊S₄, S₄×C₇⋊S₄, …
maximal normal subgroup of	#72222243, D₇×A₄⋊S₄, S₄×C₇⋊S₄, …
maximal quotient of	C₂×#72222237, #72246255

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