#72222017

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

Aliases

No aliases for this group.

Splittings

Fitting, derived, Sylow

UT(4,𝔽₃) ⋊ C₂³

Subgroups

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	UT(4,𝔽₃)	C₂³
Frattini	C₃³	D₃³
socle	C₃	((((C₃⋊D₃)⋊C₃)⋊D₃)⋊C₃)⋊C₂
LCS	G ▷ UT(4,𝔽₃)
UCS	C₁

1021 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 1021 conjugacy classes of subgroups — too many to draw. Only its 46 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,f^a=f^2,f^b=f^2g^2,f^c=f^2,f^d=fg,f^e=fh,g^a=g^2,g^c=g^2,g^e=gi,h^b=hi,h^c=h^2,h^d=hi,i^b=i^2,i^c=i^2

Conjugacy classes & character table

48 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72222023, C₂×#72222017, #72254100
maximal normal subgroup of	C₂×#72222017
maximal quotient of	C₂×#72222017
automorphism group of	C₃⋊((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, 6, 1, F.6^2);;
gap> SetConjugate(coll, 6, 2, F.6^2*F.7^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 6, 5, F.6*F.8);;
gap> SetConjugate(coll, 7, 1, F.7^2);;
gap> SetConjugate(coll, 7, 3, F.7^2);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8^2);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 9, 2, F.9^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);