#72222249

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

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#14554335 ⋊ C₂
derived	#10624759 ⋊ C₂×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, cosocle, hypocenter	#10624759	C₂×C₃
Frattini	C₃³	C₃×C₃⋊(C₃⋊D₃)
Fitting	#14554335	C₂
socle	C₃	C₃×#92338
LCS	G ▷ #10624759
UCS	C₁

1831 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 1831 conjugacy classes of subgroups — too many to draw. Only its 101 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|d^3=h^2,e^3=h^2,c^a=c^2,d^a=d^2h,d^b=dh^2,d^c=df,e^a=e^2h,e^b=eh^2,e^c=efg^2,f^a=fh,f^c=fh,f^d=fh,g^a=gh^2,g^d=gh,h^a=h^2

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72222257

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