#72220201

order	2592
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₃²×C₄

Aliases

No aliases for this group.

Splittings

derived, Sylow

C₃⁴ ⋊ C₄×C₈

Subgroups

	subgroup	G/H
center, Frattini, hypercenter	C₂	#157²
commutator, hypocenter	C₃⁴	C₄×C₈
Fitting, socle	C₂×C₃⁴	C₄²
prime radical	C₂×C₃⋊(C₃⋊(C₃⋊D₃))	C₂×C₄
cosocle	#10620150	C₂²
LCS	G ▷ C₃⁴
UCS	C₁ ◁ C₂

340 conjugacy classes of subgroups.

Full subgroup data →

Presentation

Polycyclic presentation:

2^5,3^4|a^2=c,b^2=d,c^2=e,f^a=fg,f^b=f^2g^2,f^c=f^2,f^d=f^2,g^a=fg^2,g^b=f^2g,g^c=g^2,g^d=g^2,h^a=f^2g^2h,h^b=hi,h^c=g^2h,h^d=h^2,i^a=gi,i^b=hi^2,i^c=fi,i^d=i^2

Conjugacy classes & character table

72 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72220215, #72220221, #72220222, …
maximal normal subgroup of	#72220215, #72220221, #72220222, …
maximal quotient of	C₂×#72220201

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