#72214331

order	2916
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂×C₃²

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

H(3,𝔽₃)² ⋊ C₂²

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	H(3,𝔽₃)²	C₂²
Frattini, socle	C₃²	C₃⋊D₃²
LCS	G ▷ H(3,𝔽₃)²
UCS	C₁

1275 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 1275 conjugacy classes of subgroups — too many to draw. Only its 61 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:

2^2,3^6|c^a=c^2,d^a=d^2,d^b=d^2,d^c=dg,e^a=e^2,e^b=e^2,e^c=eh,f^a=f^2,f^d=fg^2,f^e=fg,g^b=g^2,h^b=h^2

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214399, C₃×#72214331, #72228645, …
maximal normal subgroup of	#72214399, C₃×#72214331, #72228645, …
maximal quotient of	#72221933, #72221971, C₃×#72214331, …

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