#72221908

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃² · Schur multiplier (H₂) = C₃⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14554265 ⋊ C₂
derived	C₃³×C₉ ⋊ C₂×C₃²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₃³×C₉	C₂×C₃²
Frattini	C₃³	C₃²×C₃⋊D₃
Fitting	#14554265	C₂
socle	C₃²	C₃×C₃⋊((C₃⋊D₃)⋊C₃)
LCS	G ▷ C₃³×C₉
UCS	C₁

1921 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 1921 conjugacy classes of subgroups — too many to draw. Only its 89 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:

2,3^7|e^3=h,d^a=d^2,d^c=dg,e^a=e^2h^2,e^b=eh,e^c=ef,f^a=f^2,f^c=fh,g^a=g^2,h^a=h^2

Conjugacy classes & character table

99 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72221920, #72221994, C₂×#72221908
maximal normal subgroup of	#72221920, #72221994, C₂×#72221908
maximal quotient of	C₂×#72221908

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