#72214719

order	3528
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, socle	C₂²×C₇² ⋊ C₃×D₃
derived	C₇²×A₄ ⋊ C₂×C₃
Sylow	C₇² ⋊ C₃×S₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₇²×A₄	C₂×C₃
Fitting, socle	C₂²×C₇²	C₃×D₃
LCS	G ▷ C₇²×A₄
UCS	C₁

240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214783, S₄×#64424, #72222215, …
maximal normal subgroup of	S₄×#64424, #72222215, #72224540
maximal quotient of	#72214783

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