#72304160

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#8534
commutator	C₂²×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×#123	C₂⁴
socle	C₂²	C₂×#8534
prime radical	#72304143	C₂
LCS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#525 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

19451 conjugacy classes of subgroups.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 64

Polycyclic presentation:

2^11|a^2=e,c^2=hij,e^2=k,f^2=j,b^a=bg,c^a=cf,c^b=cik,d^a=djk,d^c=dhijk,e^c=ei,f^a=fij,f^b=fh,f^c=fk,f^d=fjk,f^e=fj,g^c=ghjk,g^f=gk,h^a=hk,i^a=ij

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304170
maximal normal subgroup of	#72304170

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", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 5, F.11);;
gap> SetPower(coll, 6, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.9*F.11);;
gap> SetConjugate(coll, 4, 1, F.4*F.10*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.8);;
gap> SetConjugate(coll, 6, 3, F.6*F.11);;
gap> SetConjugate(coll, 6, 4, F.6*F.10*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.8*F.10*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.11);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);