#72277433

order	4374
automorphism	?
homology	Abelianization (H₁) = C₂×C₃ · Schur multiplier (H₂) = C₁

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14550839 ⋊ C₂
derived	#10624485 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₉	#92199
commutator, cosocle, hypocenter	#10624485	C₂×C₃
Frattini	C₃²×C₉	C₃×C₃⋊D₃
Fitting	#14550839	C₂
socle	C₃	C₃×#92199
LCS	G ▷ #10624485
UCS	C₁ ◁ C₉

481 conjugacy classes of subgroups; only the 29 normal subgroups are shown.

normal subgroups only

This group has 481 conjugacy classes of subgroups — too many to draw. Only its 29 normal subgroups are shown.

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

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 81

Polycyclic presentation:

2,3^7|b^3=h^2,c^3=fh,d^3=g^2h^2,e^3=h^2,c^a=c^2f^2h^2,c^b=cfh,d^a=d^2gh,d^b=dgh,d^c=de,e^a=ef^2g^2h,e^c=ef,e^d=eg,f^a=f^2h,f^d=fh,g^a=g^2h,g^c=gh

Conjugacy classes & character table

102 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72277433
maximal normal subgroup of	C₂×#72277433
maximal quotient of	C₂×#72277433
automorphism group of	#6864

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