#72222168

order	3528
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₂×C₃

Aliases

No aliases for this group.

Splittings

Fitting, socle, derived	C₂²×C₇² ⋊ C₂×C₃²
Hall	C₂²×C₇⋊D₇ ⋊ C₃²
Sylow	C₇² ⋊ C₂×C₃×A₄

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇²	C₂×C₃²
LCS	G ▷ C₂²×C₇²
UCS	C₁

152 conjugacy classes of subgroups; only the 24 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 24 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:

2,3^2,2^2,7^2|d^b=e,d^c=de,e^b=de,e^c=d,f^a=f^6,f^b=g,f^c=f^2,g^a=g^6,g^b=f^5g^3,g^c=g^2

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	A₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇), #72222303
maximal normal subgroup of	A₄×ΓL(1,𝔽₈)⋊AGL(1,𝔽₇)
maximal quotient of	#72222303

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