#72277462

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#10624740 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₃	#11757741
commutator, Fitting, cosocle, hypocenter	#10624740	C₂²
Frattini	C₃³	D₃×C₃⋊D₃
socle	C₃²	((C₃⋊D₃)⋊C₃)⋊D₃
LCS	G ▷ #10624740
UCS	C₁ ◁ C₃

500 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 33 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) = 54

Polycyclic presentation:

2^2,3^6|c^3=h,c^a=c^2h^2,c^b=c^2h^2,d^a=dh^2,d^b=d^2,d^c=dg,e^a=e^2,e^b=e^2,e^c=ef,g^a=g^2,g^b=gh^2,g^d=gh,h^a=h^2,h^b=h^2

Conjugacy classes & character table

61 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72277465
maximal normal subgroup of	#72277465

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