#72259582

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

derived, Sylow

C₃²×H(3,𝔽₃) ⋊ C₂²×C₄

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#14519181
commutator, hypocenter	C₃²×H(3,𝔽₃)	C₂²×C₄
Frattini	C₂×C₃	D₃×#1113
Fitting, cosocle	C₂×C₃²×H(3,𝔽₃)	C₂³
socle	C₂×C₃³	C₂×D₃²
prime radical	C₂×#11758090	C₂
LCS	G ▷ C₃²×H(3,𝔽₃)
UCS	C₁ ◁ C₂

1530 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 1530 conjugacy classes of subgroups — too many to draw. Only its 99 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) = 72

Polycyclic presentation:

2^4,3^5|a^2=d,e^a=e^2,e^b=e^2,e^c=e^2,f^a=e^2f,f^b=f^2,f^c=f^2,g^b=g^2,g^e=gi,g^f=gi^2,h^a=hi,h^c=h^2,i^a=i^2,i^c=i^2

Conjugacy classes & character table

108 conjugacy classes — view the full character table.

Relationships

maximal subgroup of

#72259637

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", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 3, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.4);;
gap> SetConjugate(coll, 5, 1, F.5^2);;
gap> SetConjugate(coll, 5, 2, F.5^2);;
gap> SetConjugate(coll, 5, 3, F.5^2);;
gap> SetConjugate(coll, 6, 1, F.5^2*F.6);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 2, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 7, 6, F.7*F.9^2);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 9, 1, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);