#72246181

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72236702 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#120⋊C₃
commutator, cosocle	C₂×H(5,𝔽₂)	C₂⁴×C₃
Frattini	C₂⁴	C₂⁴×A₄
Fitting	#72236702	C₃
hypercenter	#127	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₃×H(5,𝔽₂)
LCS	G ▷ C₂×H(5,𝔽₂) ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ #127

14939 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 14939 conjugacy classes of subgroups — too many to draw. Only its 228 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) = 64

Polycyclic presentation:

2^4,3,2^6|d^2=fk,g^2=k,h^2=k,b^a=bf,d^a=dfk,d^b=dfk,d^c=dfk,g^c=gij,g^e=h,h^c=hi,h^e=ghk,h^g=hk,i^e=jk,i^g=ik,j^e=ijk,j^g=jk,j^h=jk

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72247039, #72267085, #72267089, …
maximal normal subgroup of	#72247039, #72267085, #72267089, …
maximal quotient of	C₂×#72246181

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