#72214399

order	5832
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₃)² ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₃)²	C₂³
Frattini, socle	C₃²	D₃²×C₃⋊D₃
LCS	G ▷ H(3,𝔽₃)²
UCS	C₁

2110 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 2110 conjugacy classes of subgroups — too many to draw. Only its 91 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:

2^3,3^6|d^b=d^2,e^a=e^2,e^b=e^2,e^c=e^2,e^d=eh,f^a=e^2f,f^b=f^2,f^c=f^2,f^d=fi,g^b=g^2,g^e=gh^2,g^f=gh,h^a=h^2,h^c=h^2,i^a=h^2i,i^c=i^2

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₃×#72214399, ((C₃⋊D₃)⋊C₃)⋊C₂², C₂×#72214399
maximal normal subgroup of	C₃×#72214399, ((C₃⋊D₃)⋊C₃)⋊C₂², C₂×#72214399
maximal quotient of	#72244026, #72244030, #72244038, …
holomorph of	(C₃⋊D₃)⋊C₃

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