#72243770

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

#10624752 ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	#10624752	C₂²
Frattini	C₃³	D₃×C₃⋊D₃
socle	C₃	#11757730
LCS	G ▷ #10624752
UCS	C₁

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

normal subgroups only

This group has 415 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) = 27

Polycyclic presentation:

2^2,3^6|e^3=h,c^b=c^2,d^a=d^2,d^b=d^2,d^c=df,e^a=d^2ef^2h^2,e^b=e^2h^2,e^c=eg,e^d=ef^2h^2,f^a=f^2h,f^b=fh^2,f^d=fh,f^e=fh^2,g^a=f^2gh^2,g^d=gh^2,h^b=h^2

Conjugacy classes & character table

45 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72243786, #72243790, C₃×#72243770
maximal normal subgroup of	C₃×#72243770
maximal quotient of	C₃×#72243770

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