#72244173

order	2592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₉

Aliases

No aliases for this group.

Splittings

Fitting	C₉²×C₄⋊C₄ ⋊ C₂
Sylow	C₉² ⋊ C₄×D₄

Subgroups

	subgroup	G/H
center	C₂²	C₂²×C₉⋊D₉
commutator	C₂×C₉²	C₂²×C₄
Frattini, socle	C₂²×C₃²	C₂²×C₃⋊D₃
Fitting	C₉²×C₄⋊C₄	C₂
prime radical	C₂×C₄⋊(C₉⋊D₉)	C₂
cosocle	C₂²×C₉²	C₂³
hypercenter	C₄⋊C₄	C₉⋊D₉
hypocenter	C₉²	C₄×D₄
LCS	G ▷ C₂×C₉² ▷ C₉²
UCS	C₁ ◁ C₂² ◁ C₄⋊C₄

1081 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 1081 conjugacy classes of subgroups — too many to draw. Only its 274 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) = 1296

Polycyclic presentation:

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

Conjugacy classes & character table

420 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72244213, C₄×D₄×C₉⋊D₉, #72254568
maximal normal subgroup of	#72244213, C₄×D₄×C₉⋊D₉, #72254568
maximal quotient of	#72244213

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