#72234652

order	3072
automorphism	?
homology	Abelianization (H₁) = C₂² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Splittings

Fitting

C₂⁴×H(5,𝔽₂) ⋊ D₃

Subgroups

	subgroup	G/H
center, Frattini	C₂²	C₂×#87588
commutator, cosocle	C₂×C₂²⋊#997	C₂²
Fitting	C₂⁴×H(5,𝔽₂)	D₃
socle	C₂⁴	C₂×#1020
hypercenter	C₂³	#87588
hypocenter	C₂²⋊#997	D₄
LCS	G ▷ C₂×C₂²⋊#997 ▷ C₂²⋊#997
UCS	C₁ ◁ C₂² ◁ C₂³

85188 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 85188 conjugacy classes of subgroups — too many to draw. Only its 64 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) = 96

Polycyclic presentation:

2^3,3,2^7|b^2=c,b^a=bc,d^b=d^2,e^b=f,e^d=f,f^b=e,f^d=ef,g^b=gh,g^d=h,h^d=gh,i^b=j,i^d=j,i^f=ik,j^b=i,j^d=ij,j^e=jk

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252245, #72258531, #72267121, …
maximal normal subgroup of	#72252245, #72267121, #72267141
maximal quotient of	#72256075, #72258528, #72261456

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