#72343933

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow

#72312267 ⋊ C₃

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#120⋊C₃
commutator, cosocle	C₂×#585	C₂³×C₃
Frattini	C₂³×C₄	C₂³×A₄
Fitting	#72312267	C₃
hypercenter	C₂×D₈	#120⋊C₃
hypocenter	H(5,𝔽₂)	C₃×#120
LCS	G ▷ C₂×#585 ▷ H(5,𝔽₂)
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×D₈

12358 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 12358 conjugacy classes of subgroups — too many to draw. Only its 92 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:

2^3,3,2^7|a^2=f,f^2=k,b^a=bf,c^a=ce,f^b=fk,g^a=gi,g^b=gi,g^c=gi,g^d=h,h^a=hj,h^b=hj,h^c=hj,h^d=gh,i^d=j,i^h=ik,j^d=ijk,j^g=jk

Conjugacy classes & character table

91 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343956, #72344011
maximal normal subgroup of	#72343956, #72344011

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