#72259306

order	3888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Splittings

Fitting	C₄×#6891 ⋊ C₂²
Sylow	#6891 ⋊ C₂×D₄

Subgroups

	subgroup	G/H
center	C₂	C₂×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃
commutator, cosocle	C₂×C₃²⋊C₉	C₂³×C₃
Frattini, socle	C₂×C₃²	C₂×C₃×D₃²
Fitting	C₄×#6891	C₂²
hypercenter	C₄	(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃
hypocenter	C₃²⋊C₉	C₂×C₃×D₄
LCS	G ▷ C₂×C₃²⋊C₉ ▷ C₃²⋊C₉
UCS	C₁ ◁ C₂ ◁ C₄

880 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 880 conjugacy classes of subgroups — too many to draw. Only its 86 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:

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

Conjugacy classes & character table

117 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊C₃, #72259991
maximal normal subgroup of	D₄×(((C₃⋊D₃)⋊C₉)⋊C₂)⋊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, 2, 3, 3, 3, 3]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 7, F.9);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 6, 2, F.6^2);;
gap> SetConjugate(coll, 6, 3, F.6^2);;
gap> SetConjugate(coll, 7, 3, F.7^2*F.9^2);;
gap> SetConjugate(coll, 7, 4, F.7*F.9^2);;
gap> SetConjugate(coll, 7, 6, F.7*F.8);;
gap> SetConjugate(coll, 8, 2, F.8^2);;
gap> SetConjugate(coll, 9, 3, F.9^2);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);